travelling salesman business

Explained: What is Traveling Salesman Problem (TSP)

• Last Updated: January 24, 2023

• A well-known mathematical problem called the Traveling Salesman Problem (TSP) aims to determine the shortest path between a number of places.
• Logistics, transportation, and manufacturing are just a few of the industries where the TSP is useful.
• The number of points, the form of the point set, and the algorithm employed can all have an impact on how the TSP is solved.
• Technology advancements like cloud computing and parallel processing have made it possible to solve the Traveling Salesman Problem effectively for larger and more complicated situations.

Traveling salesman problem is not new for delivery-based businesses. Its recent expansion has insisted that industry experts find optimal solutions in order to facilitate delivery operations.

The major challenge is to find the most efficient routes for performing multi-stop deliveries. Without the shortest routes, your delivery agent will take more time to reach the final destination. Sometimes problems may arise if you have multiple route options but fail to recognize the efficient one.

Eventually, traveling salesman problem would cost you time and result in late deliveries . So, before it becomes an irreparable issue for your delivery company, let us understand the traveling salesman problem and find optimal solutions in this blog.

Table of Content

• What is the Traveling Salesman Problem (TSP)?
• Commom challenges of Traveling Salesman Problem (TSP)

What are Some Popular Solutions to Traveling Salesman Problem?

What are other optimal solutions to the traveling salesman problem, what are some real-life applications of traveling salesman problem.

• How TSP and VRP Combinedly Pile up Challenges?
• Can a route planner resolve Traveling Salesman Problem (TSP)?

What is Traveling Salesman Problem (TSP)?

The traveling Salesman Problem (TSP) is a combinatorial problem that deals with finding the shortest and most efficient route to follow for reaching a list of specific destinations.

It is a common algorithmic problem in the field of delivery operations that might hamper the multiple delivery process and result in financial loss. TSP turns out when you have multiple routes available but choosing a minimum cost path is really hard for you or a traveling person.

How difficult is it to solve?

It is quite difficult to solve TSP as it is known as NP-hard, which means there is no polynomial time algorithm to solve it for more numbers of addresses. So, with an increasing amount of addresses, the complexity of solving TSP increases exponentially.

So, it is impossible to find TSP solutions manually. Also, many mathematical algorithms and the fastest computers fail to solve TSP.

However, TSP can be eliminated by determining the optimized and efficient path using approximation algorithms or automated processes.

Common Challenges of Traveling Salesman Problem (TSP)

Being a salesman is not easy, as you need to face various unavoidable challenges in your everyday schedules.

• Firstly, every day, salespeople have to carry out a number of deliveries in a very limited time, so there are a lot of time constraints. To overcome this, you need to plan your routes in a way that you make the most out of them.
• Secondly, there are chances of last-minute changes. Sometimes you get extra and urgent visits to make, while sometimes, some visits are postponed or canceled due to the customer’s unavailability.
• Lastly, a math problem, a combinatorial optimization problem, arises. A combinatorial optimization problem is a problem that is mathematically complex to solve as you have to deal with many variables.

These are major challenges in the Traveling Salesman Problem (TSP) as you are required to create a route with the shortest distances using hundreds and thousands of permutations and combinations that asks for less fuel, fulfill on-time delivery to customers, and are ready to modify routes considering last minute changes.

These are some of the near-optimal solutions to find the shortest route to a combinatorial optimization problem.

1. Nearest Neighbor (NN)

The Nearest Neighbor Method is probably the most basic TSP heuristic. The method followed by this algorithm states that the driver must start by visiting the nearest destination or closest city. Once all the cities in the loop are covered, the driver can head back to the starting point.

Solving TSP using this efficient method, requires the user to choose a city at random and then move on to the closest unvisited city and so on. Once all the cities on the map are covered, you must return to the city you started from.

2. The Branch and Bound Algorithm

The Branch & Bound method follows the technique of breaking one problem into several little chunks of problems. So it solves a series of problems. Each of these sub-problems may have multiple solutions. The solution you choose for one problem may have an effect on the solutions of subsequent sub-problems.

3. The Brute Force Algorithm

The Brute Force Approach takes into consideration all possible minimum cost permutation of routes using a dynamic programming approach. First, calculate the total number of routes. Draw and list all the possible routes that you get from the calculation. The distance of each route must be calculated and the shortest route will be the most optimal solution.

• Multi-Agent System : Involves distributing the pair of cities into groups. Then assign a single agent to discover the shortest path, covering all the cities in the assigned group.
• Zero Suffix Method : This method solves the classical symmetric TSP and was introduced by Indian researchers.
• Multi-Objective Evolutionary Algorithm : This method solves the TSP using NSGA-II
• Biogeography-based Optimization Algorithm : This method is based on the migration strategy of animals for solving optimization issues.
• Meta-Heuristic Multi Restart Iterated Local Search : This method states that the technique is more efficient compared to genetic algorithms.

Blow Away TSP using Upper

Need a permanent solution for recurring TSP? Sign up with Upper to keep your tradesmen updated all the time. Lay off your manual calculation and adopt an automated process now!

Most businesses see a rise in the Traveling Salesman Problem (TSP) due to the last mile delivery challenges . The last mile delivery is the process of delivering goods from the warehouse (or a depot) to the customer’s preferred location. Considering the supply chain management, it is the last mile deliveries that cost you a wholesome amount.

At the same time, you need to sacrifice financial loss in order to maintain your current position in the market. Suppose last mile delivery costs you $11, the customer will pay $8 and you would suffer a loss. This is because of pre-defined norms which may favor the customer to pay less amount.

This hefty last mile delivery cost is the result of a lack of Vehicle routing problem(VRP) software. VRP finds you the most efficient routes so that operational costs will not get increase. So, by using the right VRP software, you would not have to bother about TSP.

Such delivery management software uses an automated process that doesn’t need manual intervention or calculations to pick the best routes. Hence, it is the easiest way to get rid of the traveling Salesman Problem (TSP).

How TSP and VRP Combinedly Pile Up Challenges?

The traveling Salesman Problem is an optimization problem studied in graph theory and the field of operations research. In this optimization problem, the nodes or cities on the graph are all connected using direct edges or routes. The weight of each edge indicates the distance covered on the route between the two cities.

The problem is about finding an optimal route that visits each city once and returns to the starting and ending point after covering all cities once.

The TSP is often studied in a generalized version which is the Vehicle Routing Problem . It is one of the most broadly worked on problems in mathematical optimization. VRP deals with finding or creating a set of routes for reducing time, fuel, and delivery costs.

Is there any real world solution to TSP and VRP?

Many solutions for TSP and VRP are based on academics which means they are not so practical in everyday life. The reason is that many of them are just limited to perfection, but need a dynamic programming-based solution. So, if businesses really want to get rid of them, they need a TSP solver integrated with route optimization software.

The right TSP solver will help you disperse such modern challenges. It offers in-built route planning and optimization solutions in such a way that your tradesman doesn’t get stranded while delivering the parcel. Also, it is equipped with an efficient algorithm that provides true solutions to the TSP. As a result, the delivery manager can create a route plan hassle-free in a few minutes.

Can a Route Planner Resolve the Traveling Salesman Problem (TSP)?

In the general case, the Traveling Salesman Problem (TSP) involves finding the shortest optimized and possible route that includes a set of stops and returns to the starting point. The number of possible routes increases exponentially as the number of locations increases. Finding the best solution becomes difficult computationally, even for moderately sized problems.

But, Upper Route Planner, a route optimization software , is built differently. Upper has all the solutions you need when talking about TSP.

For example, if you are in charge of planning delivery routes with more than 500 stops in them, all you need to do is import an Excel or CSV file with multiple addresses into Upper, review, allot delivery drivers, optimize, and dispatch with a single click. This delivery route planning solution saves you hours of time spent on planning delivery routes and optimizing them.

Also, once the delivery is completed, Upper lets you collect proof of delivery. This is how the Upper Route Planner is a simple solution to the Traveling Salesman Problem.

Upper Route Planner

A simple-to-use route planner that every one is talking about

TSP stands for traveling Salesman Problem, while VRP is an abbreviation form of vehicle routing problem (VRP). In the delivery industry, both of them are widely known by their abbreviation form.

Yes, you can prevent TSP by using the right route planner. The online route planner helps you get the optimized path so that your delivery agents don’t have to deal with such challenges. In addition, they don’t struggle with multiple routes. Instead, they can progress on the shortest route.

The new method has made it possible to find solutions that are almost as good. This was done by the Christofides algorithm, the popular algorithm in theoretical computer science. This algorithm plugs into an alternate version of the problem that finds a combination of paths as per permutations of cities. It made the round trip route much longer. The round trip produced by the new method, while still not being efficient enough is better than the old one.

The vehicle routing problem (VRP) reduces the transportation costs as well as drivers’ expenses. It helps you serve more customers with fewer fleets and drivers. Thus, you don’t have any variation in the time taken to travel.

Create Optimized Routes Using Upper and Bid Goodbye to Traveling Salesman Problem

As a business owner, If you are dealing with TSP and want to get rid of them, we recommend using a TSP solver like Upper Route Planner. The online route planner is capable of plucking out the most efficient routes no matter how big your TSP is. It has an in-built sophisticated algorithm that helps you get the optimized path in a matter of seconds.

Therefore, you won’t fall prey to such real-world problems and perform deliveries in minimum time. Upper’s delivery route planner offers a dedicated driver app that makes sure your tradesman doesn’t go wrongfooted and quickly wraps up pending deliveries. Don’t just agree with our words, book a demo on Upper and disperse TSP once and for all.

Rakesh Patel, author of two defining books on reverse geotagging, is a trusted authority in routing and logistics. His innovative solutions at Upper Route Planner have simplified logistics for businesses across the board. A thought leader in the field, Rakesh's insights are shaping the future of modern-day logistics, making him your go-to expert for all things route optimization. Read more.

Sign Up Now!

Get weekly updates from Upper Route Planner.

Tired of Manual Routing?

Related Posts

Sales Route Planning Guide: Know How to Plan the Best Sales Routes

Sign Up with Upper Route Planner and automate your daily business process route planning, scheduling, and optimizing!

https://www.google.com/

https://www.upperinc.com/guides/travelling-salesman-problem/

www.google.com

Grab a FREE Trial of Upper

• Plan routes with hundreds of stops in a minute
• Schedule routes months in advance
• Collect reliable proof of delivery
• Track drivers live for real-time updates
• Experience unparalleled customer support

Grab a FREE Trial of Upper TODAY!

• Schedule routes in advance for weeks
• Collect proof of delivery to maintain accountability
• Experience 24/7 customer support
• Smart reporting to get real-time insights

What is a Travelling Salesman Problem (TSP)? Explained!

• Author: Diksha Bhandari
• Read Time: 9 min
• Published: September 14, 2023
• Last Update: November 8th, 2023

Table of Contents

• Asset Tracking
• Dispatch Management
• Driver Behaviour
• Electric Vehicle
• Employee Management
• Field Sales And Service
• Fleet Management
• Fuel Management
• GPS Software
• GPS Trackers and Hardware
• Last Mile Delivery
• Lead Management
• Leave And Attendance
• Roster Management
• Route Optimisation
• Route Planning
• Sensor Integration
• Task and Workflow
• Tech and Beyond
• TrackoBit Industries
• TrackoField
• TrackoField Industries
• TrackoMile Industries
• Vehicle Tracking
• Video telematics

Want to know what a travelling salesman problem (TSP) is? Need solutions to real-life TSP challenges. Learn here. 

Do you also look for the shortest route on Google Maps before embarking on a trip?

I am sure, you know multiple routes to reach the office, the mall, or your desired location, but checking on the internet before leaving home has become a ritual. It becomes all the more important to refer to maps when you have to pick up friends or colleagues along the way.

‘ADD STOPS’

Yes, you are right! 💯

That’s what solving the TSP challenge using software means!

What is a Travelling Salesman Problem (TSP)?

The traveling salesman problem is the popular combinatorial optimisation challenge in mathematics and computer science. The prime objective of the problem is to determine the shortest possible route a salesperson must take to cover a set of locations in one go and then return to the starting point.

Addressing travelling salesman challenges and their optimisation are more relevant in this time and age, especially in the supply chain, logistics and delivery space.

TSP may result in delayed deliveries and slimming profits as it’s not easy for delivery agents to choose the most viable and cost-effective route in real-time.

What are Traveling Salesman Problem Challenges to Solve?

When a salesperson is in the field hopping from one client site to another, finding out the best and the shortest route is an added pressure on the agent. In today’s day and age, distance isn’t the only factor that defines efficiency. There are several factors, such as time, fuel consumption, capacity, etc. that together define efficiency.

However, addressing the travelling salesman challenges involves mitigating a few unavoidable challenges along the way that field agents face themselves.

1. Time Constraints

Sales agents often have a tight schedule with multiple deliveries to make with a short TAT. Similarly, in TSP, optimising routes to minimise travel time is a fundamental challenge.

2. Last-minute Changes

Eleventh-hour changes are not a new concept for salespeople. They encounter urgent visits and last-minute cancellations a lot. Similarly, TSP solutions must be adaptable to handle dynamic scenarios and route modifications.

3. Resource Efficiency

Just as salespersons aim at reducing fuel costs and ensuring on-time deliveries, TSP solutions such as TrackoMile must strive for resource optimisation by reducing travel distances and delivery TAT.

4. Objective Diversification

While solving the travelling salesman problem (TSP) , optimising multiple objectives such as cost, time, and environmental factors adds complexity as solutions need to balance conflicting goals.

5. Combinatorial Complexity

TSP is a combinatorial optimisation problem, which means it involves complicated mathematical calculations with numerous variables. Sometimes, complex scenarios get further intricate as multiple variables are involved.

6. Adaptability and Scalability

Similarly, how sales agents adjust to the routes on the fly, the route algorithm must be flexible and responsive to real-time changes such as spiking volumes, vehicle breakdown or traffic slow down. A TSP solution must have a good appetite to handle large datasets and complex networks.

Also Read 4 Key Solutions for Fuel Management System 2023

Top 5 Solutions to The Travelling Salesman Problem

The traveling salesman problem solutions offer various trade-offs between computational intricacies and the quality of the resolution, allowing practitioners to choose the best-suited approach based on their needs and problems.

Here are the Top 5 solutions to the Traveling Salesman Problem (TSP) :

1. Brute Force Algorithm

The Brute Force algorithm is a straight approach to solving the Traveling Salesman Problem (TSP). It systematically explores all possible routes to identify the shortest one among them all. While it guarantees an optimal solution, its downside lies in its major time complexity, making it practical only for small TSP challenges.

2. Nearest Neighbour Algorithm

The Nearest Neighbour method is the simplest heuristic for the TSP. It starts from the first location and repeatedly selects the closest unvisited location to form a tour. Although it is quick to implement this method, it may always yield the optimal solution for it prioritises proximity over other factors.

3. Genetic Algorithm

This technique or method draws inspiration from nature itself. They evolve TSP solutions through selection, crossovers and mutation. They pick the best routes and mix them up. This creates new routes that might be even better. Then, they keep the best ones and repeat the mixing and picking process. Survival of the fittest in the true sense.

4. Ant Colony Optimisation (ACO)

Ants have a tendency to leave pheromones on the shorter routes they find, calling fellow ants on the same route. They keep leaving more pheromones on the shorter routes they find. Over time, the collective behaviour of the ants causes them to converge on the shortest route. Inspired by the nature of ants, ACO finds the shortest route by analysing the trails of data left by artificial ants based on the strength of these data trails.

5. Dynamic Programming

Dynamic Programming is like solving a puzzle, step-by-step, by breaking it into smaller pieces. In TSP challenges, it finds the best route to visit all locations. It begins with figuring out the shortest route between two locations; then it builds on that to find ways to more locations. It’s a smart TSP solution for small scenarios but may require significant memory resources for larger and more complex problems.

What Are Real-world Travelling Salesman Problem Applications?

The Traveling Salesman Problem (TSP) has a wide array of applications across various domains due to its relevance in optimising routes and sequences. Here are several crucial real-word TSP applications and implementations in the real world.

1. TSP implementation in Logistics and Delivery Services

The logistics and supply chain sectors have the widest TSP applications.

• Courier, Express & Parcel : Companies like FedEx, UPS, and DHL rely on TSP algorithms to optimise delivery routes for their fleet of delivery trucks. By finding the most efficient sequence of stops, they minimise fuel consumption , reduce delivery TAT, and save on operational overheads too.
• On-demand Delivery : Food delivery companies, instant grocery delivery apps and at-home appointment platforms like Swiggy, BlinkIt and UrbanCompany, respectively, leverage TSP solutions to ensure timely delivery. Enhancing the customer experience and increasing the number of deliveries each rider can make.

2. TSP Applications in Transportation and Urban Planning Waste collection routes, Traffic light synchronisation, optic cable installation, etc. are some areas where TSP Solutions works like a knight in shining armour. Other real-world TSP applications include

• Public Transport : City planners and public transport agencies use TSP principles to design bus, tram and train routes that reduce travel for passengers.
• Emergency Service Dispatch : Ambulance services, Police PCR vans employ TSP algorithms to dispatch vehicles quickly and efficiently in response to emergency calls. Finding the shortest route to reach the incident location can save lives.
• Urban Mobility Solution : In the era of ride-sharing and on-demand mobility apps like Uber, Ola, Lyft, etc., real-world TSP applications become prominent. TSP solutions optimise the route to destinations, ensuring quick and cost-effective transportation.

Other significant real-life applications of the Travelling Salesman Problem are

• TSP in Healthcare and Medical Research – for DNA sequencing and understanding genetic patterns and diseases.
• TSP in Manufacturing and Production – In circuit board manufacturing and job scheduling of technicians.
• TSP in Robotics and Autonomous Vehicles -Self-driving cars and drones use TSP-like algorithms for efficient navigation.

Solving the Travelling Salesman Problem – Last Mile Delivery Route Optimisation

Route optimisation is the key to efficient last-mile delivery . In order to attain flawless route optimisation, the software must solve the traveling salesman problem every step of the way.

Why it’s essential to solve TSP for Last Mile Delivery?

In simple and minimal words, solving TSP problems helps in many ways:

• Saves Time : It makes deliveries faster, so your customers get orders sooner.
• Customer Satisfaction : Fast deliveries give you an edge over the competition and enhance customer experience too.
• Saves Money : It reduces fuel wastage and vehicle wear, making deliveries cheaper.
• Environment Friendly : It lowers pollution by using fewer vehicles and shorter routes.
• Happy Staff: Drivers and dispatchers have less stress and can finish their work faster.

How do we solve the travelling salesman problem for last-mile delivery?

Solving TSP challenges for Last-mile delivery is like solving a big jigsaw puzzle. There are a hundred thousand addresses to visit daily. The software must find the shortest and most optimised route to them and come back to the starting point at the end.

• Our route optimisation software , TrackoMile, leverages capacity management , routing algorithms and robust rule engines to present the most optimal combination of delivery addresses. Thereby giving the most optimally planned routes or trips.
• All delivery managers have to do is upload the CSV file of the addresses or integrate TrackoMile to their CRM to fetch the delivery addresses. Now trip allocation, route optimisation, dispatch and everything else happen in a few clicks.
• ETA when the delivery is en route, POD when the order is delivered successfully, and trip analysis, are added features to simplify overall operations.

The Vehicle Routing Problem is very similar to TSP, with wide applications in logistics, delivery services and transportation. While TSP focuses on finding the shortest route for a single traveller visiting various locations, VRP deals with multiple vehicles serving multiple customers, considering added constraints like vehicle capacity, TATs and more.

How Can AI Help in Solving Traveling Salesman Problem (TSP)?

AI or Artificial Intelligence are becoming the driving force for business growth across various industrial sectors. AI particularly aids in solving the Traveling Salesman Problem(TSP) in the logistics and delivery sector by employing advanced algorithms and techniques. What are a few tricks up AI’s sleeves that help in automating TSP resolution? Let’s find out!

1. Advanced Algorithms

AI algorithms such as Genetic Algorithms, ACO, simulated annealing and a few others mentioned above, tackle complex Travelling Salesman Problem scenarios.

2. Machine Learning

Gathering information from historical data and optimising routes based on real-time insights is what AI is best for. Machine learning models are trained to adapt to changing conditions, like traffic, weather and delivery constraints, to provide a more accurate plan of action.

3. Parallel Computing

AIi enables the use of a parallel computing process, which means solving multiple segments of TSP simultaneously. This accelerates the problem-solving process for large-scale challenges.

4. Heuristic Improvement

TSP Heuristics powered by AI can groom initial solutions, gradually improving their results over time. These heuristics can be applied iteratively by AI to reach better results.

5. Hybrid Approaches

Applying hybrid algorithms is not a new technique to refine techniques and produce more accurate results. AI on top of it singles out data-oriented combinations that work the best in varied use cases.

Wrapping Up!

The travelling salesman problem’s importance lies in its real-world applications. Whether optimising delivery routes, planning manufacturing processes or organising circuit board drilling, finding the most efficient way to cover multiple locations is crucial to minimise costs and save time.

The TSP problems have evolved over the years, and so have TSP algorithms, heuristics and solutions. With the advent of advanced technologies such as GPS and machine learning, TSP continues to adapt and find new applications in emerging fields, cementing its status as a fundamental problem in optimization theory and a valuable tool for various industries. Mobility automation software like Trackobit, TrackoMile and TrackoField resort to TSP heuristics to solve challenges along the way.

Read Blog – Best Delivery Route Planner Apps for 2023

Traveling Salesman Problem FAQs

What is tsp.

TSP is an abbreviation for Traveling Salesman Problem. It’s the routing problem that deals with finding the shortest route to travel to a combination of locations in the most optimal manner.

Is Travelling Salesman Problem Solvable?

Yes, the Traveling Salesman Problem (TSP) is solvable, but the time to find the solution can grow proportionately with an increase in the number of locations. With the evolution of travelling salesman problem applications, various TSP algorithms and heuristics, their hybrids have also emerged.

Wh at is the objective of TSP?

The objective of the Traveling Salesman Problem (TSP) is to find the shortest possible route that covers all given locations or waypoints and comes back to the starting point with the least resource utilisation.

Diksha Bhandari

Currently creating SaaSy content strategies for TrackoBit and TrackoField, this content professional has dedicated a decade of her life to enriching her portfolio and continues to do so. In addition to playing with words and spilling SaaS, she has a passion for traveling to the mountains and sipping on adrak wali chai.

• Author Showcase
• All Blog Post

Never Miss a Beat

Related Blog

Top 12 Google Map Alternatives – Offering Precise Navigation

Explore our best picks for 12 free Google map alternatives that offer accurate and secure location and navigational solutions. 

Food Delivery Trends to Watch in 2024 & Beyond

Food and technology are both revolving along with consumer demands. Here are some of the food delivery trends to watch in 2024.

Biofuel – An Alternative to Fossil Fuels in Transportation | G20

Transportation industries are moving towards a greener future through the sustainable use of biofuels like Ethanol — discussed in the G20 2023 summit.

Top 10 Navigation Apps for Android and iOS | 2024

Discover the 10 best navigation apps in 2024 that offer personalised navigation just like Sherpas (experienced guides of Himalayan terrain) do.

What are Thematic Maps? Types, Applications And Advantages

Maps have come a long way, and their usage has gone beyond just navigating distances. Thematic maps are now being used to interpret data, gain information, etc. 

From a Bachelor Pad to a Multinational Firm: Tracking TrackoBit’s 5-year Journey

“It’s when ordinary people rise above the expectations and seize the opportunity that milestones truly are reached.” – Mike Huckabee

Stay Updated on tech, telematics and mobility. Don't miss out on the latest in the industry.

Thank you for reaching out! We'll speak to you soon.

In the meantime, why not find out more about us, explore our products, or visit our blog?

Cookie Consent

We use cookies to enhance and personalize your browsing experience. By continuing to use our website, you agree to our Privacy Policy.

Route Optimization

Traveling salesman problem (tsp) and how tech can solve it.

Mar 2, 2020

11 mins read

Updated: Mar 9, 2023

Salespersons are required to travel frequently as a part of their job, with the aim of maximizing their business opportunities. However, their daily schedules are often subject to sudden changes due to scheduled and last-minute customer appointments, making it difficult for them to plan optimal travel routes. As a result, salespersons are forced to take longer routes to reach their destinations, which can result in missed appointments and lost business opportunities.

What is the Traveling Salesman Problem (TSP)? 

The Traveling Salesman Problem (TSP) involves finding the shortest possible route to multiple destinations and returning to the starting point. However, this is a complex task due to various constraints such as traffic, last-minute customer requests, and strict delivery windows. Successfully solving the TSP challenge can optimize supply chains and reduce logistics costs, making it an important problem to tackle.

Despite its seemingly simple definition, the TSP becomes increasingly difficult as more cities are added to the itinerary. For example, when delivering to six cities, there could be up to 360 possible routes to consider. Finding the most efficient and feasible path requires evaluating every possible route, which is a computationally challenging task. This has been an ongoing problem since the 1990s.

The TSP has relevance in various industries, such as astronomy, computer science, car navigation, agri-business, airline crew scheduling, computer-generated art, internet planning, logistics planning and scheduling, and microchip production.

What is the objective of TSP?

The problem statement for the Traveling Salesman Problem (TSP) has remained the same over the years: “Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city and returns to the origin city?” This was one of the most well-formulated optimizations in 1930 and remains a powerful solution for logistics-related problems. Variations in TSP problems have since made a disruptive difference in many other areas, including DNA sequencing.

TSP is no longer just about the problem of a salesman traveling to multiple cities in the least amount of time, using the least amount of money, while traveling the shortest distance. It now involves the world gearing towards the Fourth Industrial Revolution.

As cutting-edge technology is being integrated into manufacturing, TSP provides access to a wide variety of operations. For instance, a TSP variation could help design an efficient microchip. Similarly, an asymmetric version of TSP can optimize the set-up cost of a paint manufacturing plant by rearranging the vicinity of colors.

Is the Traveling Salesman Problem solvable?

Considering the basic problem itself, optimizing the logistics of a traveling salesman is under more monetary pressure than ever in history. Even if the problem is kept within city limits, it would be significant enough to become news. This is a real problem at a time when convenience is preferred over everything. A country’s GDP depends on its e-commerce purchases, which is something TSP enthusiasts didn’t foresee five decades ago.

India’s e-commerce industry is expected to contribute $300 billion by 2030, and the entire burden of it rests on the backseat of a delivery bike or transport. There are significant costs associated with this, particularly with return logistics being a pain point. What complicates the TSP logic further is the ground reality where the whole market is dependent upon cash-on-delivery, making it impossible to determine if a delivery will earn its cost back at times.

The problem also manifests itself in ride-sharing. Those who carpool to work heavily depend on affordable and comfortable transportation, and the twice-a-day practice that occurs during rush hour is not easy on anyone. There is added time and pressure for picking up each passenger, constant deviation from the main route to drop off others, and stress associated with it cannot be optimized.

Why is the Traveling Salesman Problem crucial to solve?

The Traveling Salesman Problem is a serious challenge for the logistics and supply chain industry because it involves optimizing the delivery routes for multiple destinations while considering various constraints such as traffic, delivery windows, and customer requests.

With multiple vehicles, more cities and multiple sales professionals, TSP becomes more challenging to solve. Not solving the TSP makes it difficult for sales professionals to efficiently reach their customers, and lead to a fall in business revenues.

Solving traveling salesman problems benefit businesses in numerous ways:

• Saves fuel and reduces the hours traveled by the sales and field professionals 
• Minimizes the distance traveled, thereby reducing the carbon footprint considerably
• Timely meetings with clients and on-time delivery of goods 
• Elevate last-mile customer experience for delivery and field service businesses

Why is Traveling Salesman Problem challenging to solve? 

In theory, solving the Traveling Salesman Problem (TSP) is relatively simple as it involves finding the shortest route for every trip within a city. However, as the number of cities increases, manually solving TSP becomes increasingly difficult. With just 10 cities, the permutations and combinations are already numerous. Adding just five more cities can exponentially increase the number of possible solutions.

It’s impossible to find an algorithm that solves every single TSP problem. There are also some constraints that make TSP more challenging to solve:

• No automated records of scheduled and last-minute business appointments 
• Traffic congestion
• Sudden change of routes 
• Increasing number of last-minute business appointments 
• Stricter customer time windows 
• Rising operational fleet costs

All these constraints give a clear insight that TSP is a real-world problem. It is impossible to solve this challenge with even the best of your manual efforts. Thus, it is necessary to harness the power of technology to manage the TSP problem efficiently.

Using the Hungarian method to solve the Travelling Salesman Problem

To feel the intensity of the problem closely, one can try solving it using the Hungarian method. If one assumes that a delivery executive makes 15 stops in a day, it would still be too large to optimize by hand. In the ride-sharing example where the car has to make four stops—picking up three passengers and returning to the point of origin—costs need to be allocated between rides. It is important not to repeat the same route twice and to solve the TSP to arrive at the minimum cost.

The calculation may seem quite simple at the moment, but scaling the execution of such optimized routes to handle deliveries worth $300 billion is a huge challenge. It would not only overwhelm one’s mind but quite possibly the RAM of one’s computer too. It seems unfair that such a large computational problem, which has taken decades to solve, has suddenly become so large that only a few have the power to tackle it.

How does Artificial Intelligence technology help in solving the Traveling Salesman Problem? 

Modern enterprises want to attract customers by providing superior service. A customer that interacts with an enterprise for a business opportunity wants the business to value their needs. 

Most customers make business decisions based on how salespeople from an enterprise respond to their requirements. The crucial aspect that they look out for in a salesperson is whether they are able to be on time for business appointments. A salesperson that travels to multiple locations should ensure they attend business appointments on time.

The Traveling Salesman Problem makes it way more difficult for salespersons to make their business appointments on time and reach their sales targets. With a lot of complexities involved in routes traveled, it is necessary for them to employ the right technology to counter them. Artificial Intelligence (AI) plays an active role in helping businesses counter route inefficiencies and solve TSP.

AI combines human intuition with complex mathematics in real-time. It analyzes a massive amount of data clearly and quickly. It mainly helps a modern enterprise to make operational, strategic and tactical decisions.

Here’s how AI solved TSP in many modern enterprises. 

Makes optimized decisions for each vehicle and route

With increasing business appointments, it would be difficult for enterprise businesses to control operational fleet costs . It is highly challenging to manage crucial business appointments without the help of technology, especially when the business handles multiple vehicles and multiple salespersons.

AI technology helps operation managers in a business to prepare and assign optimal sales schedules for their workforce. By taking into account vicinity, vehicle capacity, customer time window, skills of sales professionals, driver skills and so on, it generates optimal appointment schedules. These optimal schedules help the businesses strictly adhere to the Service Line of Agreement (SLA)

Read:      The Definitive Guide to Logistics Route Optimization

Saves fuel and labor costs

According to the “An Analysis of the Operational Costs of Trucking: 2022 Update”, the average fuel cost per mile decreased further from $0.308 in 2020 to $0.296 in 2021.

Since the outbreak of the Covid-19 pandemic in 2020, fuel costs have been at record low levels. However, as the world moves towards a post-pandemic stage in 2023, fuel costs have started to rise steadily. The Ukraine-Russia war in 2022 and its trade shocks have had a significant impact on fuel prices, resulting in a sharp increase. As a result, fuel costs have become a larger proportion of operational costs for carriers in 2023.

The increasing number of appointments on a daily basis for salespersons makes it difficult for businesses to minimize fuel costs.

Employing the right technology like AI will help salespersons complete the maximum number of customer appointments by traveling a minimal distance. Its optimally allocated route plans reduce the traveling time and help businesses avoid wastage of fuel and labor costs.

Recognize the right addresses

When you are in a rush to complete your sales visits, unclear addresses can induce delays. If the driver is going to ride in an unfamiliar area, delays will increase. These delays can even cause missed appointments.

By using an AI application like route optimization software , enterprise businesses can avoid delays caused due to incorrect addresses. Its competent geocoding capabilities convert unclear or incomplete text addresses into geographical coordinates , thereby reducing the chances of reaching wrong customer locations.

Reduces cost per mile

According to the “An Analysis of the Operational Costs of Trucking: 2022 Update” by the American Transport Research Institute, the average cost per mile of trucks continued to decrease from $1.646 in 2020 to $1.585 in 2021.

As fuel costs continue to rise and the world moves towards a post-pandemic period, there is a potential for an increase in the average cost per mile for trucks. This poses a challenge for modern enterprise businesses who want their salespersons to reach their targets without increasing operational fleet costs. To achieve their sales targets, salespersons need to attend more business appointments.

AI technology with its algorithms factors business goals and delivery constraints to come up with efficient routes. Its optimal route recommendations enable salespersons to take the cost-minimizing route, thereby attending a maximum number of business appointments in a day and minimizing cost per mile. 

Improves productivity 

What happens when salespersons attend business appointments in locations where they may not have many opportunities? It results in deadhead or empty miles. Empty miles kill the productivity of the fleet, and drive up operational costs considerably. 

AI technology coupled with data insights helps modern enterprise businesses analyze their past performance of sales appointments. It helps them identify business opportunities that incur huge costs and work out new ways to counter them. Based on historical data, it assists stakeholders of a business to rightsize their fleet requirements and direct them to locations where higher demand is expected. The past data records and predictive capabilities of AI help them implement strategies that improve productivity of their sales professionals.

With modern enterprise businesses struggling from incorrect routes and delays, it has become highly-crucial to solve TSP situations. Scientists have been working on solving the TSP problem for decades. But the modern AI-backed routing software has gone closer towards solving traveling salesman problems with their complex algorithms and enhanced route planning capabilities.

The Locus Dispatch Management Platform is the best-in-class tool that has helped many enterprise businesses solve their traveling salesman problems. Its algorithms help sales professionals generate the most efficient schedules for any complicated routes without any manual interference. By employing its AI capabilities, you can increase the number of appointments with your customers by traveling a minimal distance, thereby making sales visits more optimal.

To build the most efficient routes for your sales schedules and counter TSP effectively, try a demo with Locus now!

Related Tags:

Healthcare / Pharmaceutical

Route optimization for the pharmaceutical industry: countering last-mile delivery problems.

Feb 28, 2020

Pharmaceutical organizations spend on average 6% of their revenue on logistics- World Health Organization and Parenteral Drug Association statistics (WHO and PDA).  The pharmaceutical industry has been successful in creating life-saving drugs through advanced research and clinical trials. Transportation and storage determine the major success of pharma products globally. A small mistake in logistics can […]

CEP and 3PL

The future of e-commerce lies in achieving same-day and slot-based deliveries.

Mar 4, 2020

Convenience. That’s the word of our times.  With all the apps doing the rounds, everything happens in an instant. Have a thought that the world should know? Type it, tweet it. Have a photo that you took on your weekend cycling trip, upload and post it with a crazy caption on Instagram. Hungry? Food from […]

• Schedule a Demo

MOST POPULAR

The Rise of Hub and Spoke Distribution Model in Modern Supply Chains

Shweta Sarma

Apr 24, 2020

How to Calculate the Cost of Transport

Jun 9, 2023

EDITOR’S PICKS

Top 4 Mid-Mile Logistics Challenges and How to Solve Them 

Dec 7, 2023

Navigating Returns: Five Strategies for Shippers to Reduce Return to Origin

Prateek Shetty

Nov 28, 2023

Last Mile Automation: Intelligent Routing is Key to Success

Mar 28, 2023

SUBSCRIBE TO OUR NEWSLETTER

Stay up to date with the latest marketing, sales, and service tips and news

Discover more from Locus Blog

Subscribe now to keep reading and get access to the full archive.

Type your email…

Continue reading

Try RoadWarrior free for 7 days

Solving the Traveling Salesman Problem

Get home early with RoadWarrior.

Enter your stops, optimize your routes, manage your team – quickly and efficiently.

Imagine a salesman who needs to visit multiple cities, but he wants to minimize the distance traveled and return to the starting point. This classic problem, known as the Traveling Salesman Problem (TSP), has been a subject of study for over a century. The applications of TSP are widespread, from logistics and agriculture to astronomy and computer-generated art. In this blog post, we will delve into the world of the Traveling Salesman Problem, exploring its history, algorithms, and real-world applications.

Short Summary

• The Traveling Salesman Problem is a complex problem of finding an optimal route for a round trip.
• Solutions to the TSP include NP-hard classification, brute force approach, dynamic programming and Christofides’ Algorithm.
• Route planning software such as OptimoRoute provide businesses with efficient tools for tackling the TSP, thereby improving their operations and saving resources.

Understanding the Traveling Salesman Problem

The Traveling Salesman Problem (TSP) is a problem of determining the most efficient route for a round trip, with the objective of maintaining the minimum cost and distance traveled. It serves as a foundational problem to test the limits of efficient computation in theoretical computer science.

The salesman’s objective in the TSP is to find a minimum weight Hamiltonian cycle, which maintains both the travel costs and the distance traveled at a minimum.

Theoretical Background

The TSP is classified as an NP-hard problem. This shows that the number of solution sequences grows rapidly with the number of cities. The brute force approach to resolving the TSP involves examining each round-trip route to ascertain the shortest one. However, as the number of cities increases, the number of round-trips to check can quickly surpass the capability of the most powerful computers. This limitation has led to the development of more sophisticated algorithms to tackle the TSP, such as dynamic programming and approximation algorithms.

The Hamiltonian Cycle Problem, also known as the Hamiltonian cycle problem, inquires whether there exists a closed walk in a graph that visits each vertex precisely once and is closely related to the TSP. Both problems have been studied extensively by computer scientists due to their implications in complexity theory and the P versus NP problem.

Optimal vs. Approximate Solutions

In solving the TSP, there is a distinction between optimal solutions and approximate solutions. Optimal solutions are the most advantageous route, while approximate solutions are round-trip routes whose lengths approach that of the most advantageous route. The brute force approach involves determining every potential solution and subsequently selecting the most advantageous one. However, this method is computationally intractable for large TSP instances.

One notable approximate solution is Christofides’ algorithm, which begins by determining the shortest spanning tree and subsequently converting it into a round-trip route. This algorithm guarantees a response that is, at most, 1.5 times the optimal solution. While not always yielding the optimal solution, approximate algorithms like Christofides’ algorithm provide a more feasible approach to solving the TSP.

Evolution of TSP Algorithms

TSP algorithms have been in existence since the 1950s, when the initial brute force approach was formulated. Subsequently, more sophisticated techniques such as dynamic programming and Christofides’ algorithm have been developed. These advanced algorithms have enabled researchers and practitioners to find near-optimal solutions to the TSP more quickly and efficiently.

By leveraging these algorithms, it is possible to solve the TSP in a fraction of the time.

Brute Force Approach

The brute force approach to TSP involves attempting all potential solutions, making it the most time-consuming and expensive method. As the number of destinations increases, the number of roundtrips likewise increases exponentially, rendering it computationally intractable even for the most powerful computers. Therefore, the brute force approach is not considered to be a viable solution for large TSP instances.

Despite its limitations, the brute force approach serves an important role in the history of TSP algorithms. It represents the initial attempt to solve the TSP and laid the groundwork for the development of more advanced and efficient algorithms.

Dynamic Programming

Dynamic programming is a technique employed to address intricate issues by segmenting them into more manageable subproblems and resolving them one at a time. It is regularly utilized to resolve the Traveling Salesman Problem due to its ability to avoid redundant calculations and identify the shortest route that visits all cities exactly once. The dynamic programming approach is more scalable than the brute force approach, as it can be employed to solve problems of any size.

However, dynamic programming is not without its limitations. It can be computationally expensive and time-consuming, as it necessitates solving a substantial number of subproblems. Despite these drawbacks, dynamic programming remains a popular and effective method for solving the TSP.

Christofides’ Algorithm

Christofides’ algorithm is an algorithm for obtaining approximate solutions to the Traveling Salesman Problem. It involves constructing a minimum spanning tree and then discovering a minimum-weight perfect matching on the odd-degree vertices of the tree. This algorithm, developed in the 1970s, utilizes a combination of graph theory and heuristics to address the TSP.

The significance of Christofides’ algorithm lies in its ability to yield routes that are guaranteed to be no more than 50 percent longer than the shortest route. While it may not always provide the optimal solution, its development marked a substantial improvement in the pursuit of efficient TSP algorithms.

Recent Advances in TSP Algorithms

In recent years, computer scientists have made significant advancements in TSP algorithms. These breakthroughs include:

• Approximation algorithms
• Metaheuristic approaches
• Local search operators
• Anytime Automatic Algorithm Selection

These cutting-edge algorithms have enabled researchers to find even more efficient solutions to the TSP, further demonstrating the importance of continued research and development in this area.

Geometry of Polynomials

Oveis Gharan and Nathan Klein used the geometry of polynomials approach to solve the TSP by representing the problem as a polynomial with variables corresponding to the edges between all the cities. This approach permits the utilization of geometric techniques and algorithms to discover an optimal or approximate solution to the problem, including determining the starting and ending point of the route.

This innovative method showcases the potential for leveraging mathematical techniques and geometrical properties in solving the TSP. As research progresses, it is expected that even more efficient algorithms and approaches will be developed, further pushing the boundaries of our understanding of the TSP.

Fractional Solutions and Rounding Techniques

Amin Saberi and Arash Asadpour developed a general rounding technique that employs randomness in an attempt to select a whole-number solution that preserves as many characteristics of the fractional solution as possible. The use of fractional solutions and rounding techniques can be utilized to construct effective approximation algorithms for the TSP.

Saberi, Gharan, and Singh implemented this general rounding technique to formulate a new approximation algorithm for the TSP. As computer scientists continue to explore new approaches and techniques, it is likely that even more powerful and efficient algorithms will be developed to tackle the complex TSP.

Practical Applications of TSP Solutions

TSP solutions are employed in a variety of industries, including logistics, astronomy, agriculture, and vehicle routing. Its applications range from planning efficient delivery routes and optimizing telescope trajectories to designing microchips and creating computer-generated art.

The widespread use of TSP solutions underscores the importance of continued research and development in this area, as advancements in TSP algorithms have the potential to positively impact numerous sectors.

Route Optimization

Route optimization is the process of determining the most efficient routes for various applications. In logistics, for example, TSP solutions can assist in enhancing efficiency in the last mile. By employing optimization techniques, businesses can reduce the number of stops, minimize total distance traveled, and ultimately save on fuel and labor costs.

The Vehicle Routing Problem (VRP), a generalized form of the TSP, focuses on discovering the most efficient set of routes or paths for multiple vehicles and hundreds of delivery locations. By utilizing TSP solutions and optimization techniques, companies can greatly improve their overall efficiency and productivity, leading to significant cost savings and improved customer service.

Last-Mile Delivery Challenges

The last-mile delivery problem refers to the final stage of a supply chain. It involves transporting goods from a transportation hub, such as a depot or warehouse, to the ultimate recipient. TSP solutions play a crucial role in addressing this challenge by optimizing delivery routes, reducing the number of stops, and minimizing total distance traveled.

For example, the Traveling Salesman Problem with Time Windows (TSPTW) approach considers specific time constraints for each delivery location, ensuring that deliveries are made within a specified time frame. By employing TSP algorithms and optimization techniques, businesses can:

• Overcome last-mile delivery challenges
• Improve overall efficiency
• Achieve cost savings
• Enhance customer satisfaction

TSP Solvers for Real-World Problems

Modern TSP solvers use advanced algorithms to provide near-optimal solutions quickly. These solvers include the routingpy library, real-life TSP and VRP solvers, and state-of-the-art TSP solvers based on local search.

By leveraging these powerful algorithms, businesses and researchers can tackle real-world TSP problems with greater efficiency and accuracy.

Branch and Bound Method

The branch and bound method is an algorithm utilized to address optimization problems, such as the TSP. It functions by exhaustively examining all potential solutions and identifying the most advantageous one. By dividing the problem into smaller subproblems and determining the optimal solution for each subproblem, the branch and bound method permits a more efficient and precise resolution to the problem.

Although the branch and bound method is computationally expensive, it has several advantages, such as being relatively straightforward to implement and capable of addressing large-scale problems. Its application in solving real-life TSP instances showcases its potential in effectively optimizing routes and minimizing costs.

Nearest Neighbor Method

The Nearest Neighbor algorithm is a greedy algorithm that finds the closest unvisited node and adds it to the sequencing until all nodes are included in the tour. While it rarely yields the optimal solution, particularly for large and intricate instances, it can be utilized effectively as a means to generate an initial feasible solution quickly.

This initial solution can then be supplied into a more sophisticated local search algorithm for further optimization. The Nearest Neighbor algorithm demonstrates that even relatively simple algorithms can play a valuable role in providing quick and feasible solutions to real-world TSP problems.

Route Planning Software for TSP

Utilizing route planning software to solve TSP problems in various industries offers numerous benefits. These software solutions, such as Route4Me, leverage advanced algorithms like Dijkstra’s Algorithm to quickly identify the most efficient route for a team.

By implementing route planning software, businesses can improve efficiency, reduce costs, and enhance customer satisfaction.

Advantages of Route Planning Software

Route planning software can help businesses in the following ways:

• Optimize routes
• Decrease the number of stops
• Minimize the total distance traveled
• Increase efficiency and productivity
• Reduce fuel and labor costs by optimizing routes and minimizing the time spent on route planning and decision-making.

Improved customer service is another benefit of route planning software. By optimizing routes and ensuring timely deliveries, businesses can enhance their reputation and build customer loyalty. In an increasingly competitive market, utilizing route planning software can give businesses a critical edge in delivering exceptional service and maintaining customer satisfaction.

Examples of Route Planning Solutions

There are numerous route planning solutions available for addressing the TSP, including vehicle routing software, optimization algorithms, and Excel sheets with order details and addresses. These solutions offer businesses a variety of options to choose from, depending on their specific needs and requirements.

Some examples of popular route planning software include OptimoRoute and Straightaway. These software solutions can help businesses tackle TSP problems efficiently, saving time and resources while improving overall operations. By leveraging advanced algorithms and powerful tools, route planning software provides a valuable solution for businesses looking to optimize their logistics and delivery processes.

Throughout this blog post, we have explored the fascinating world of the Traveling Salesman Problem, delving into its history, algorithms, and practical applications. From the early brute force approach to modern approximation algorithms and route planning software, the TSP continues to challenge and inspire researchers, computer scientists, and businesses alike. As we continue to push the boundaries of our understanding of the TSP, the potential for new and innovative solutions to real-world problems remains vast and exciting.

Frequently Asked Questions

Has anyone solved the traveling salesman problem.

No one has successfully come up with an algorithm to efficiently solve every traveling salesman problem, despite notable progress being made over the years.

What is the Traveling Salesman Problem (TSP)?

The Traveling Salesman Problem is an optimization problem that seeks to determine the most efficient route for a round trip, with the aim of minimizing cost and distance traveled.

What are some examples of industries that benefit from TSP solutions?

TSP solutions are widely used in industries such as logistics, astronomy, agriculture, and vehicle routing, providing a range of benefits.

What is the difference between optimal and approximate solutions in TSP?

Optimal solutions in TSP provide the most advantageous route, while approximate solutions offer a similar route whose length is close to that of the optimal solution.

These solutions can be used to solve a variety of problems, such as finding the shortest route between two cities or the most efficient way to deliver goods. They can also be used to optimize the use of resources, such as time and money.

What is an example of a modern TSP solver?

Routingpy is an example of a modern TSP solver, providing comprehensive tools to address the Traveling Salesman Problem and Vehicle Routing Problem.

It offers a range of features, including an intuitive user interface, fast computation times, and a wide range of optimization algorithms. It also provides a comprehensive set of tools for analyzing and visualizing the results of the optimization process.

Related Articles

Android route planning simplified: the best apps on the market.

7 minute read

Best Route Planners for iPhone: Navigating the App Choices

Route4Me: A Detailed Look at Features and Pricing

Solving The Traveling Salesman Problem (TSP) For Deliveries

The traveling salesman problem is a classic optimization problem that businesses still face today. The problem is simple: given a list of cities, find the shortest route that visits each city exactly once and then returns to the starting point. Despite its simplicity, the traveling salesman problem solution is hard to find.

Even with modern computing power, the best algorithms can only find approximate solutions for large instances of the problem. Nevertheless, the traveling salesman problem remains an essential tool for businesses because it can be used to model many real-world problems, such as vehicle routing and resource allocation.

Why Is the TSP Important for Businesses

What businesses use tsp, how to solve the travelling salesman problem, how to choose the best method for your business needs, tips for implementing a tsp solution for your business.

Businesses rely on efficient operations to be successful. They want to minimize costs while still being able to provide a good or service that meets the needs of their customers. The traveling salesman problem is a classic example of how businesses can optimize their operations.

This might not seem like a significant problem. Still, it turns out that finding the optimal solution to the traveling salesman problem can have a substantial impact on a business’s bottom line.

For example, consider a company that delivers packages to different locations. The faster they can make their deliveries, the more packages they can deliver overall, and the more money they will make. Using an algorithm to solve the traveling salesman problem, the company can find the shortest possible route between the different delivery locations and thus make their deliveries more efficient .

Another example is a company that provides transportation services, such as a taxi or Uber. If the company can find the shortest trip route between all pick-up and drop-off locations, they can save on fuel costs and reduce the amount of time their drivers are on the road. This can significantly decrease operating expenses, which can be passed on to the customer at lower prices.

The traveling salesman problem can also be applied to other types of businesses, such as manufacturing. For instance, if a company has a factory with multiple machines, they need to find the shortest path that goes through all of the machines to minimize the amount of time and energy required to produce their product.

In recent years, businesses have begun to use the traveling salesman problem to optimize their operations .

UPS, for example, uses the traveling salesman problem to plan its delivery routes. The company has a large fleet of vehicles and needs to find the most efficient way to deliver packages to its customers.

To do this, UPS created a particular piece of software called ORION ( On-Road Integrated Optimization and Navigation ). This software uses GPS sensors to find the optimal route for each delivery. In addition, the software considers things like traffic, weather, and construction to find the best possible route.

By using the traveling salesman problem, UPS has reduced the distance its drivers travel by millions of miles each year, which has saved the company millions of dollars in fuel costs.

Amazon is a master of logistics, and a big part of that is due to their use of the traveling salesman problem. Amazon uses TSP to optimize its shipping routes, ensuring that each package travels the shortest distance possible while still hitting every Amazon warehouse.

This helps save time and money, and it means that your package will arrive as quickly as possible. Amazon has even filed a patent for an algorithm that explicitly addresses the traveling salesman problem, showing how important this issue is to their business.

Another major shipping company, DHL, also uses the traveling salesman problem to optimize its routes. DHL has a similar system to Amazon, using an algorithm to find the shortest distance between all of its warehouses. This helps DHL save time and money on each delivery, which can be passed on to the customer at lower prices.

There are various methods that businesses can use to solve the traveling salesman problem.

The Brute Force Method

One of the simplest methods for solving the traveling salesman problem is the brute force method. This method involves trying every possible route and selecting the shortest one.

While this method is reasonably straightforward, it can be very time-consuming for businesses with many locations. In addition, the brute force method can become very complicated very quickly, making it difficult for companies to implement.

The Branch and Bound Method

Another popular method for solving the traveling salesman problem is the branch and bound method. This method involves breaking the problem down into smaller sub-problems, which can be solved more easily.

The branch and bound method is very effective for businesses with many locations, as it can help simplify the problem. In addition, the branch and bound method can be used to find approximate solutions, which can save businesses time and money. The disadvantage to this method is that it can be challenging to implement, requiring a lot of planning and coordination.

The Genetic Algorithm

The genetic method involves using algorithms to simulate the process of natural selection. The genetic method can eventually converge on an optimized solution by starting with a large pool of potential solutions and then slowly eliminating the least fit ones.

There are several benefits to using this approach. First, it is very efficient, often finding much better solutions than those found by other methods. Second, it is highly parallelizable, meaning that it can take advantage of modern computer hardware to find solutions even faster.

However, there are some drawbacks to using the genetic method as well. First, it can be challenging to understand and implement. Second, it can be computationally expensive, meaning that businesses may need to invest in more powerful computer hardware to take advantage of this method.

The Linear Programming Method

One of the most popular methods for solving the traveling salesman problem is the linear programming method. This method involves solving linear equations to find the optimal solution.

There are both advantages and disadvantages to using linear programming to solve the traveling salesman problem. On the positive side, linear programming is relatively easy to implement and can often find reasonably good solutions in a reasonable amount of time.

On the negative side, linear programming can sometimes produce very poor results, especially if the data is not well-suited to this type of analysis. In addition, linear programming can be pretty slow for large problem sizes.

Start Using RouteManager!

Finding the optimal solution to the TSP is tricky, and even small instances can take years to solve. As a result, many businesses rely on heuristic methods to find suitable solutions quickly. However, there are a few essential considerations when choosing a heuristic method for solving the TSP.

Size of the Problem

The first consideration is the size of the problem. For small instances, with a few dozen locations, any methods described above could potentially find a good solution. However, for large instances, with hundreds or even thousands of sites, it is crucial to choose a method that can scale well.

The genetic algorithm and the branch and bound method can scale well to large problem sizes. As a result, these methods are often the best choice for businesses with many locations.

Time Constraints

The second consideration is time constraints. For many businesses, it is crucial to find a solution quickly, even if the solution is not perfect. In these cases, heuristic methods are often the best choice.

The genetic algorithm and the linear programming method can find solutions quickly. As a result, these methods are often the best choice for businesses that need to find a solution in a short amount of time.

Computational Resources

The third consideration is computational resources. For many businesses, the cost of investing in more powerful computer hardware can be prohibitive. Therefore, it is vital to choose a method that requires less computational power in these cases.

The genetic algorithm and the linear programming method are both relatively computationally intensive. As a result, these methods may not be the best choice for businesses with limited computational resources.

Once you have selected a method for solving the TSP, you can do a few things to improve the chances of finding a good solution.

One of the most important things you can do is ensure that your data is accurate and up-to-date. This is especially important if you are using a heuristic method, as even minor errors in the data can lead to suboptimal solutions.

Another essential thing to consider is the parameters of your chosen method. For example, the genetic algorithm uses several parameters that can be adjusted to impact the algorithm’s performance. It is essential to experiment with different parameter values to find the best combination for your data and your business needs.

Solving the Traveling Salesman Problem (TSP) is essential for businesses that need to optimize delivery routes. Several different methods can be used to solve the TSP, and the appropriate method will depend on the specific needs of your business.

Nicole Fevrin, Senior Product Marketing Manager, has been with WorkWave for over four years. She works on the Route Manager, GPS, and ServMan products. Nicole has over 21 years of experience in B2B and B2C Marketing in various industries and possess a Master’s Degree in Communication Studies. Her background industries range from ultra-luxury and cosmetics to commodities and home services. This range has afforded her a well-rounded perspective of customer insights and various business models.

Related Posts

How to Optimize Your Customer Experience With Enhanced Optimized Routing

7 Reasons Why Last-Mile Delivery Is Critical for Supply Chain Success

How to Hire a Delivery Driver: A Step-by-Step Guide

What Is A Travelling Salesman Problem (TSP)?

Published Date: March 2, 2024

Table of Contents

The majority of field salespeople get up in the morning only having a general notion of what to do to make the best possible use of their workday. They want to reach multiple customers, as well as others along their journey that they are unaware of, however they might not follow a well-planned timetable. This weakness results in either retracing or getting out of time. The lack of sales route planning delays client meetings and missed opportunities.

Consider a salesperson who wants to go to several cities, but desires to cover as few miles as possible and come back to where they started. The Travelling Salesman Problem (TSP), a widely recognized problem, has remained the subject of research for more than a century. Finding the lowest possible weight Hamiltonian cycle that keeps both the travel expenses and the amount of distance travelled to a minimum is the salesman’s goal in the TSP.

The task of determining the shortest possible path or route for the salesman to travel provided a starting point, several cities (nodes), as well as a finishing point, is known as the “Travelling Salesman Problem” (TSP). This is a typical computational problem in fieldwork that could cause monetary damage and disrupt several field processes. 

TSP occurs when there are several routes accessible but it is very difficult for you or the traveller to select the one with the lowest cost. By utilizing route optimization techniques to identify more profitable routes, field businesses can lower the release of greenhouse gases by travelling shorter distances.

What Is The History Of Travelling Salesman Problem?

The Travelling Salesman Problem, or TSP, has its roots at the beginning of the nineteenth century, but Merrill M. Flood was the first person to formalize it mathematically in 1930 while trying to figure out a school bus routing problem. The term “TSP” was first used in a 1949 RAND Corporation paper. It was then used by Hassler Whitney at Princeton under the title of the “48 states problem,” which meant the shortest path was examined for travelling all of the surrounding 48 United States.

The challenge of a salesperson travelling to several locations in the shortest amount of time, spending the smallest sum of money, and covering the shortest distance is only one aspect of TSP. The world is currently preparing for the fourth industrial revolution that will result in it. As advanced technology is included in the manufacturing process, TSP offers access to a broad range of functions. For example, a form of TSP could aid in the construction of an effective microprocessor. Analogously, by changing the location of colors, an asymmetric form of TSP can optimize the cost of setting up of a paint production unit.

Major Challenges Of Travelling Salesman Problem

When salespeople wake up, nearly every one of them considers how to best utilize their workday. They have a great deal of planned appointments as well as many impromptu ones with their clients. Being a salesperson is challenging since your daily routine will inevitably provide you with a number of obstacles. Some of the travelling salesmen challenges are:

• First of all, salesmen are under constant time limitations as they complete numerous deliveries in a condensed amount of time each day. They must arrange their routes to maximize their potential in order to get around this.
• The work schedules of salesmen are typically flexible. This defect commonly forces salespeople to go back and forth between their steps and take longer routes in order to get to client locations. This absence of planning for the journey leads to lost meetings and chances.
• The salesperson nowadays faces road congestion, last-minute client demands, and tight delivery window timings along with the travel challenges that beset those before them.
• Parallel to how salespeople modify their routes on the spot, the route algorithm needs to be adaptable and quick to sudden occurrences like traffic jams, automobile failures, and volume spikes. A TSP solution needs to be able to deal with complicated networks and huge databases with ease.
• The number of routes grows dramatically in tandem with the number of destinations. The computing power of even the most advanced systems is surpassed by this hyperbolic development. 

5 Most Reliable And Successful Solutions For The Travelling Salesman Problem

Since the first brute force technique was developed in the 1950s, TSP techniques have remained in use. Further refined methods like nearest neighbor algorithms and dynamic programming have been created since then. With the aid of all of these sophisticated algorithms, researchers and developers can now solve the TSP almost with the same speed and efficiency as they did before. One can solve the TSP in a much less amount of time by utilizing these algorithms.

The Brute Force Approach

In order to find the shortest distinct solution, the Brute Force method, sometimes referred to as the Naive Method, computes and contrasts each potential set of paths or routes. One has to first figure out the overall amount of routes, then sketch and list every feasible route in order to resolve the TSP employing the Brute-Force method. The best course of action is to determine each route’s distance and then select the one that is the shortest. This is rarely helpful outside of conceptual computing seminars and is only practical for small issues.

The Nearest Neighbour Approach

The most basic way to resolve the TSP is with this approach. This approach usually starts from the closest location, visiting all other locations, and then returning to the initial location. The following are the steps needed to solve the TSP employing the Nearest Neighbour approach:

• Select an arbitrary location.
• Find the closest unexplored location and get there.
• The salesperson needs to go back to the starting point after visiting each location.

The Branch And Bound Approach

Determining the least cost for the rest of the routes that the beginning stages of methods such as the brute-force approach give is the focus of the branch and bound approach. Also known as Parallel TSP, this approach makes the assumption that we don’t wish to attempt every “possible” path. Even applying logic would likely help to simplify it. 

We won’t, for instance, travel to the drop-off location that is the longest distance before we head back to the location that is closest to the warehouse. With the use of nodes and “costs” associated with each node, the approach calculates the likelihood that a particular option will result in a problem-solving solution.

Dynamic Programming

Dynamic programming is a method for handling intricate issues by breaking them down into smaller, easier-to-handle subproblems and addressing each one separately. It can find the shortest path that covers each city precisely once and avoids doing duplicate computations, which is why it is frequently used to solve the Travelling Salesman Problem. Because the method of dynamic programming can be utilized for solving issues of any size, it is more versatile than the brute force approach. Dynamic programming does have certain restrictions, though. It can be time-consuming and operationally costly because it requires the solution of a large number of smaller issues.

Other Optimal Solutions

• The multi-agent system divides the two cities into clusters. Next, designate one agent to find the shortest route that passes via each of the cities in the designated cluster.
• The Zero Suffix Method was developed by Indian researchers and resolves the traditional symmetric TSP.
• Multi-Objective Evolutionary Method: This technique uses NSGA-II to resolve the TSP.
• Biogeography-based Optimisation Algorithm: This technique is for resolving optimization problems on how animals migrate.
• Meta-Heuristic Multi Restart Iterated Local Search: This approach claims to be more effective than genetic search algorithms.

What Are Real-Time Travelling Salesman Problem Applications?

A widely recognized optimization problem with many practical applications across many industries is the travelling salesman problem (TSP). Every one of these practical uses exemplifies the TSP’s adaptability and the various challenges it has been used to tackle. Following are some of the TSP’s more noteworthy uses:

Vehicle Routing:  Delivery and distribution route optimization is a common application of the TSP. Locations in this application correlate to delivery sites, and the salesperson is symbolized by a delivery vehicle. The idea is to determine the shortest path that makes precisely one stop at each destination before returning to the initial location. This issue is crucial in the world of delivery firms because it allows them to minimize fuel use, lower the number of vehicles needed, and guarantee the on-time distribution of products.

Logistics And Manufacturing: In the transportation and logistics sector, where businesses must optimize their supply routes to lower expenses and boost efficacy, the TSP is a prevalent issue. For instance, in order to save petrol and delivery time, a delivery service could have to determine the shortest path that stops at every one of its clients in a certain location.

The TSP is also applicable to production and manufacturing scheduling, as these fields require businesses to optimize their manufacturing processes to save expenses and boost productivity. For instance, in order to reduce downtime and increase production, a manufacturing facility may need to determine the quickest path to all of the machinery that requires maintenance.

Network Design And Optimization : In network architecture and optimization, the TSP is employed by businesses to optimize the flow of data, products, or services among a network of nodes. For instance, in order to reduce the loss of signal and enhance network efficiency, a telecommunications provider might need to determine the shortest path between its network nodes.

Circuit Design: The TSP is utilized in the area of electronics to create effective circuit schematics. In this case, the salesperson is the electrical signal that must pass via each city precisely once, and the cities are the circuit’s constituent parts. The objective is to determine the shortest path that minimizes the entire circuit’s resistance and impedance.

Robotics And Automation:  The TSP also has applications in the fields of automation and robotics, where businesses must maximize machine or robot motion in order to cut expenses and boost productivity. For instance, in order to do different duties like choosing and arranging products or examining machinery, a robot might be required to determine the shortest route to take to every site in a plant.

How TSP Solves Last-Mile Delivery Problems?

The last step in a supply chain is the focus of the last-mile delivery issue. It entails moving cargo from a central point of transportation, like a terminal or warehouse, to the final destination. 

• Inadequate Route Planning
• Delivery Windows
• Traffic Congestions
• High Travel/Fuel Costs
• Late/Untimely Deliveries

By streamlining routes for delivery, cutting down on stops, and minimizing the total kilometers travelled, TSP solutions are essential in tackling this problem. The Travelling Salesman Problem yields effective solutions that optimize and reduce last-mile delivery costs. These technologies assist delivery companies in identifying a series of courses or routes that minimize delivery expenses. A number of delivery destinations, warehouse locations, and delivery automobiles are all part of the delivery issue arena.

How AI Helps In Solving Travelling Salesman Problem (TSP)?

Artificial intelligence (AI) methods including neural networks, genetic algorithms, and ant colony optimization were all used to solve the TSP effectively. Supply chain management and logistics have benefited greatly from AI. This involves enhancing transportation routes or reducing the amount of miles driven when delivering products. The use of artificial intelligence assists business operational leaders in creating and allocating the best possible sales plans for their employees. It provides the best possible meeting plans by accounting for factors such as proximity, automobile capacity, client time window, salespeople’s abilities, driver skills, and many more. 

We can anticipate significantly more advanced applications in TSP solutions given the swift development in AI. This might entail the creation of novel artificial intelligence methods that can deal with bigger and more complicated problems or the application of quantum algorithms for faster resolution. The development of AI has revolutionized the way we approach and deal with these issues, increasing productivity and effectiveness across a range of businesses. We do not doubt that AI will play a substantially greater role in resolving TSP and related issues as the technology develops.

The Travelling Salesman Problem is a complex statistical puzzle that simulates the daily activities of a salesperson. It describes the endeavors of a door-to-door salesperson who is attempting to determine the most efficient and/or fastest manner to complete all of the locations on their list of visits in a specific amount of time (often a workday). Initially, it used to be a salesperson’s issue, but far more people deal with it nowadays. 

Field workers can learn about optimization concepts and use them when dealing with a range of real-world issues through exploring the TSP. Even though the TSP is still a difficult issue, advances in computer technology and algorithms have made it easier to get to and solve. In the end, solving the TSP along with other challenging optimization issues can enhance both our personal life and the world that we dwell in.

Related Blogs

Top 20 Sales Blogs To Follow In 2024

Top 10 Sales Conferences For Salespersons To Attend In 2024

A Detailed Brief On Daily Sales Reports

10 Typical Sales Commission Structures with Formulas and Examples

Inside Sales Vs Outside Sales

A Definitive Guide To SaaS Sales In 2023

A Detailed Brief About Sales Performance Management

10 Best Sales Podcasts For Inside And Outside Reps

12 Effective Sales Appointment Setting Tips

How to increase sales of a new product in field sales?

How to Create Steps for an Automated Field Sales Funnel

How to classify and identify potential leads

7 Best Sales Mapping Software For Your Business

How To Prepare Effectively For A Client Sales Meetings?

A complete guide on how AI helps your Sales Business to grow?

A Brief Guide On Soft Selling And Its Techniques

A Detailed Brief On The Buyer’s Journey In Sales

How to setup B2B Sales funnels And Strategies

10 Sales Conferences Every Salesperson Should Attend In 2023

How To Write Sales Follow-Up Emails After No Response With Sample Email Templates

Important Field Sales KPIs Every Sales Manager Should Know

What Is Sales Data: Importance And Types Of Sales Data Reports 

A Brief Knowledge On Sales Analysis: How To Perform An Effective Sales Analysis?

7 Constructive Field Sales Canvassing Tips For 2024

How to set up a successful sales management process?

8 Productive Tips For Field Sales Reps To Generate More Leads

10 Best Ways To Boost Employee Engagement And Motivation

7 Best Advanced Sales Training Program To Upgrade Your Sales Career

How To Prepare Yourself For A Sales Supervisor Job Interview

12 Best AI Tools to Increase the Sales of Your Business

8 Best Practices For Keeping An Organized Sales Pipeline

10 Books That Well Help You To Become A Better Salesperson

Top 20 sales blogs to follow in 2023

How Can Salespeople Improve Their Chances Of Getting An Appointment?

A complete guide on solar sales- Tips to improve solar sales

How Does A Blog Directly Impact The Business Of A Company

How to resolve the conflict between your Sales and Marketing Teams

A Brief Guide To Sales Methodology: Best Sales Methodologies For Your Team

How to handle objections in sales

10 Best Sales Enablement Tools In 2023

How to use the right tone and tonality in sales.

10 best sales closing techniques for salespeople

How to plan a commission structure for your salespeople

How to plan a 30 60 90 sales plan with an example

Importance of Regression Analysis in sales forecasting

Importance of Open-Ended Questions in sales

How to Plan for a New Sales Territory

A Complete Guide On Door-To-Door Sales: Tips For Effective Door-To-Door Selling

How to Improve customer service mindset for your Staff

What Is Sales Volume And How To Use Sales Volume Effectively To Get Revenue

5 Best Task Management Software For Field Employees

How To Set Sales Targets For Your Team: 6 Sales Target Tips To Follow

How To Make A Sales Team More Productive: 8 Tips To Follow

How To Boost Sales For A Small Business In 2022

10 Tips to Increase Your Collection Agent Performance

Top 25 Team Motivational Quotes For Better Productivity

8 Essential Team Management Skills Every Manager Must Know

How To Track Sales Person Of Your Organization

How To Prioritize Your Leads Or Prospects In Sales

5 Most Common Myths Related To Employee Management

How To Stay Focused At Work: 6 Ways That Can Increase Field Employee Productivity

5 Best Ways To Start A Good Sales Conversation With Any Prospect

Why Social Selling Is Important After The Pandemic

Why Setting Up Goals In Work Is Important

5 Types Of Sales Pitches To Be Known By Every Salesperson

A Look Through Into Customer Journey & Its Significance In Sales

Effective Tips For Sales Reps To Plan A Productive Sales Week

The Importance of Route Optimization for the Sales Team

Virtual Vs Face to Face Meetings: Which is Effective in Sales?

How Field Sales Differ From Inside Sales

7 Tips to Increase Sales for Your Small Business

Join our newsletter today to receive instant notifications when we publish new articles and have other updates.

Traveling salesman problem

This web page is a duplicate of https://optimization.mccormick.northwestern.edu/index.php/Traveling_salesman_problems

Author: Jessica Yu (ChE 345 Spring 2014)

Steward: Dajun Yue, Fengqi You

The traveling salesman problem (TSP) is a widely studied combinatorial optimization problem, which, given a set of cities and a cost to travel from one city to another, seeks to identify the tour that will allow a salesman to visit each city only once, starting and ending in the same city, at the minimum cost. 1

• 2.1 Graph Theory
• 2.2 Classifications of the TSP
• 2.3 Variations of the TSP
• 3.1 aTSP ILP Formulation
• 3.2 sTSP ILP Formulation
• 4.1 Exact algorithms
• 4.2.1 Tour construction procedures
• 4.2.2 Tour improvement procedures
• 5 Applications
• 7 References

The origins of the traveling salesman problem are obscure; it is mentioned in an 1832 manual for traveling salesman, which included example tours of 45 German cities but gave no mathematical consideration. 2 W. R. Hamilton and Thomas Kirkman devised mathematical formulations of the problem in the 1800s. 2

It is believed that the general form was first studied by Karl Menger in Vienna and Harvard in the 1930s. 2,3

Hassler Whitney, who was working on his Ph.D. research at Harvard when Menger was a visiting lecturer, is believed to have posed the problem of finding the shortest route between the 48 states of the United States during either his 1931-1932 or 1934 seminar talks. 2 There is also uncertainty surrounding the individual who coined the name “traveling salesman problem” for Whitney’s problem. 2

The problem became increasingly popular in the 1950s and 1960s. Notably, George Dantzig, Delber R. Fulkerson, and Selmer M. Johnson at the RAND Corporation in Santa Monica, California solved the 48 state problem by formulating it as a linear programming problem. 2 The methods described in the paper set the foundation for future work in combinatorial optimization, especially highlighting the importance of cutting planes. 2,4

In the early 1970s, the concept of P vs. NP problems created buzz in the theoretical computer science community. In 1972, Richard Karp demonstrated that the Hamiltonian cycle problem was NP-complete, implying that the traveling salesman problem was NP-hard. 4

Increasingly sophisticated codes led to rapid increases in the sizes of the traveling salesman problems solved. Dantzig, Fulkerson, and Johnson had solved a 48 city instance of the problem in 1954. 5 Martin Grötechel more than doubled this 23 years later, solving a 120 city instance in 1977. 5 Enoch Crowder and Manfred W. Padberg again more than doubled this in just 3 years, with a 318 city solution. 5

In 1987, rapid improvements were made, culminating in a 2,392 city solution by Padberg and Giovanni Rinaldi. In the following two decades, David L. Appelgate, Robert E. Bixby, Vasek Chvátal, & William J. Cook led the cutting edge, solving a 7,397 city instance in 1994 up to the current largest solved problem of 24,978 cities in 2004. 5

Description

Graph theory.

In the context of the traveling salesman problem, the verticies correspond to cities and the edges correspond to the path between those cities. When modeled as a complete graph, paths that do not exist between cities can be modeled as edges of very large cost without loss of generality. 6 Minimizing the sum of the costs for Hamiltonian cycle is equivalent to identifying the shortest path in which each city is visiting only once.

Classifications of the TSP

The TRP can be divided into two classes depending on the nature of the cost matrix. 3,6

• Applies when the distance between cities is the same in both directions

• Applies when there are differences in distances (e.g. one-way streets)

An ATSP can be formulated as an STSP by doubling the number of nodes. 6

Variations of the TSP

Formulation

The objective function is then given by

To ensure that the result is a valid tour, several contraints must be added. 1,3

There are several other formulations for the subtour elimnation contraint, including circuit packing contraints, MTZ constraints, and network flow constraints.

aTSP ILP Formulation

The integer linear programming formulation for an aTSP is given by

sTSP ILP Formulation

The symmetric case is a special case of the asymmetric case and the above formulation is valid. 3, 6 The integer linear programming formulation for an sTSP is given by

Exact algorithms

Branch-and-bound algorithms are commonly used to find solutions for TSPs. 7 The ILP is first relaxed and solved as an LP using the Simplex method, then feasibility is regained by enumeration of the integer variables. 7

Other exact solution methods include the cutting plane method and branch-and-cut. 8

Heuristic algorithms

Given that the TSP is an NP-hard problem, heuristic algorithms are commonly used to give a approximate solutions that are good, though not necessarily optimal. The algorithms do not guarantee an optimal solution, but gives near-optimal solutions in reasonable computational time. 3 The Held-Karp lower bound can be calculated and used to judge the performance of a heuristic algorithm. 3

There are two general heuristic classifications 7 :

• Tour construction procedures where a solution is gradually built by adding a new vertex at each step
• Tour improvement procedures where a feasbile solution is improved upon by performing various exchanges

The best methods tend to be composite algorithms that combine these features. 7

Tour construction procedures

Tour improvement procedures

Applications

The importance of the traveling salesman problem is two fold. First its ubiquity as a platform for the study of general methods than can then be applied to a variety of other discrete optimization problems. 5 Second is its diverse range of applications, in fields including mathematics, computer science, genetics, and engineering. 5,6

Suppose a Northwestern student, who lives in Foster-Walker , has to accomplish the following tasks:

• Drop off a homework set at Tech
• Work out a SPAC
• Complete a group project at Annenberg

Distances between buildings can be found using Google Maps. Note that there is particularly strong western wind and walking east takes 1.5 times as long.

It is the middle of winter and the student wants to spend the least possible time walking. Determine the path the student should take in order to minimize walking time, starting and ending at Foster-Walker.

Start with the cost matrix (with altered distances taken into account):

Method 1: Complete Enumeration

All possible paths are considered and the path of least cost is the optimal solution. Note that this method is only feasible given the small size of the problem.

From inspection, we see that Path 4 is the shortest. So, the student should walk 2.28 miles in the following order: Foster-Walker → Annenberg → SPAC → Tech → Foster-Walker

Method 2: Nearest neighbor

Starting from Foster-Walker, the next building is simply the closest building that has not yet been visited. With only four nodes, this can be done by inspection:

• Smallest distance is from Foster-Walker is to Annenberg
• Smallest distance from Annenberg is to Tech
• Smallest distance from Tech is to Annenberg ( creates a subtour, therefore skip )
• Next smallest distance from Tech is to Foster-Walker ( creates a subtour, therefore skip )
• Next smallest distance from Tech is to SPAC
• Smallest distance from SPAC is to Annenberg ( creates a subtour, therefore skip )
• Next smallest distance from SPAC is to Tech ( creates a subtour, therefore skip )
• Next smallest distance from SPAC is to Foster-Walker

So, the student would walk 2.54 miles in the following order: Foster-Walker → Annenberg → Tech → SPAC → Foster-Walker

Method 3: Greedy

With this method, the shortest paths that do not create a subtour are selected until a complete tour is created.

• Smallest distance is Annenberg → Tech
• Next smallest is SPAC → Annenberg
• Next smallest is Tech → Annenberg ( creates a subtour, therefore skip )
• Next smallest is Anneberg → Foster-Walker ( creates a subtour, therefore skip )
• Next smallest is SPAC → Tech ( creates a subtour, therefore skip )
• Next smallest is Tech → Foster-Walker
• Next smallest is Annenberg → SPAC ( creates a subtour, therefore skip )
• Next smallest is Foster-Walker → Annenberg ( creates a subtour, therefore skip )
• Next smallest is Tech → SPAC ( creates a subtour, therefore skip )
• Next smallest is Foster-Walker → Tech ( creates a subtour, therefore skip )
• Next smallest is SPAC → Foster-Walker ( creates a subtour, therefore skip )
• Next smallest is Foster-Walker → SPAC

So, the student would walk 2.40 miles in the following order: Foster-Walker → SPAC → Annenberg → Tech → Foster-Walker

As we can see in the figure to the right, the heuristic methods did not give the optimal solution. That is not to say that heuristics can never give the optimal solution, just that it is not guaranteed.

Both the optimal and the nearest neighbor algorithms suggest that Annenberg is the optimal first building to visit. However, the optimal solution then goes to SPAC, while both heuristic methods suggest Tech. This is in part due to the large cost of SPAC → Foster-Walker. The heuristic algorithms cannot take this future cost into account, and therefore fall into that local optimum.

We note that the nearest neighbor and greedy algorithms give solutions that are 11.4% and 5.3%, respectively, above the optimal solution. In the scale of this problem, this corresponds to fractions of a mile. We also note that neither heuristic gave the worst case result, Foster-Walker → SPAC → Tech → Annenberg → Foster-Walker.

Only tour building heuristics were used. Combined with a tour improvement algorithm (such as 2-opt or simulated annealing), we imagine that we may be able to locate solutions that are closer to the optimum.

The exact algorithm used was complete enumeration, but we note that this is impractical even for 7 nodes (6! or 720 different possibilities). Commonly, the problem would be formulated and solved as an ILP to obtain exact solutions.

• Vanderbei, R. J. (2001). Linear programming: Foundations and extensions (2nd ed.). Boston: Kluwer Academic.
• Schrijver, A. (n.d.). On the history of combinatorial optimization (till 1960).
• Matai, R., Singh, S., & Lal, M. (2010). Traveling salesman problem: An overview of applications, formulations, and solution approaches. In D. Davendra (Ed.), Traveling Salesman Problem, Theory and Applications . InTech.
• Junger, M., Liebling, T., Naddef, D., Nemhauser, G., Pulleyblank, W., Reinelt, G., Rinaldi, G., & Wolsey, L. (Eds.). (2009). 50 years of integer programming, 1958-2008: The early years and state-of-the-art surveys . Heidelberg: Springer.
• Cook, W. (2007). History of the TSP. The Traveling Salesman Problem . Retrieved from http://www.math.uwaterloo.ca/tsp/history/index.htm
• Punnen, A. P. (2002). The traveling salesman problem: Applications, formulations and variations. In G. Gutin & A. P. Punnen (Eds.), The Traveling Salesman Problem and its Variations . Netherlands: Kluwer Academic Publishers.
• Laporte, G. (1992). The traveling salesman problem: An overview of exact and approximate algorithms. European Journal of Operational Research, 59 (2), 231–247.
• Goyal, S. (n.d.). A suvey on travlling salesman problem.

Navigation menu

Algorithms for the Travelling Salesman Problem

The Traveling Salesman Problem (TSP) is the challenge of finding the shortest path or shortest route for a salesperson to take, given a starting point, a number of cities (nodes), and optionally an ending point. It is a well-known algorithmic problem in the fields of computer science and operations research, with important real-world applications for logistics and delivery businesses.

There are obviously a lot of different routes to choose from, but finding the best one—the one that will require the least distance or cost—is what mathematicians and computer scientists have spent decades trying to solve.

It’s much more than just an academic problem. Finding more efficient routes using route optimization algorithms increases profitability for delivery businesses, and reduces greenhouse gas emissions because it means less distance traveled.

In theoretical computer science, the TSP has commanded so much attention because it’s so easy to describe yet so difficult to solve. The TSP is known to be a combinatorial optimization problem that’s an NP-hard problem, which means that the number of possible solution sequences grows exponential with the number of cities. Computer scientists have not found any algorithm that can solve this problem in polynomial time, and therefore rely on approximation algorithms to try numerous permutations and select the shortest route with the minimum cost.

The main problem can be solved by calculating every permutation using a brute force approach and selecting the optimal solution. However, as the number of destinations increases, the corresponding number of roundtrips grows exponentially and surpasses the capabilities of even the fastest computers. With 10 destinations, there can be more than 300,000 roundtrip permutations. With 15 destinations, the number of possible routes could exceed 87 billion.

For larger real-world travelling salesman problems, when manual methods such as Google Maps Route Planner or Excel route planner no longer suffice, businesses rely on approximate solutions that are sufficiently optimized by using fast tsp algorithms that rely on heuristics. Finding the exact optimal solution using dynamic programming is usually not practical for large problems.

Three popular Travelling Salesman Problem Algorithms

Here are some of the most popular solutions to the Travelling Salesman Problem:

1. The brute-force approach

The Brute Force approach, also known as the Naive Approach, calculates and compares all possible permutations of routes or paths to determine the shortest unique solution. To solve the TSP using the Brute-Force approach, you must calculate the total number of routes and then draw and list all the possible routes. Calculate the distance of each route and then choose the shortest one—this is the optimal solution. 

This is only feasible for small problems, rarely useful beyond theoretical computer science tutorials.

2. The branch and bound method

The branch and bound algorithm starts by creating an initial route, typically from the starting point to the first node in a set of cities. Then, it systematically explores different permutations to extend the route one node at a time. Each time a new node is added, the algorithm calculates the current path's length and compares it to the optimal route found so far. If the current path is already longer than the optimal route, it "bounds" or prunes that branch of the exploration, as it would not lead to a more optimal solution.

This pruning is the key to making the algorithm efficient. By discarding unpromising paths, the search space is narrowed down, and the algorithm can focus on exploring only the most promising paths. The process continues until all possible routes are explored, and the shortest one is identified as the optimal solution to the traveling salesman problem. Branch and bound is an effective greedy approach for tackling NP-hard optimization problems like the travelling salesman problem.

3. The nearest neighbor method

To implement the Nearest Neighbor algorithm, we begin at a randomly selected starting point. From there, we find the closest unvisited node and add it to the sequencing. Then, we move to the next node and repeat the process of finding the nearest unvisited node until all nodes are included in the tour. Finally, we return to the starting city to complete the cycle.

While the Nearest Neighbor approach is relatively easy to understand and quick to execute, it rarely finds the optimal solution for the traveling salesperson problem. It can be significantly longer than the optimal route, especially for large and complex instances. Nonetheless, the Nearest Neighbor algorithm serves as a good starting point for tackling the travelling salesman problem and can be useful when a quick and reasonably good solution is needed.

This greedy algorithm can be used effectively as a way to generate an initial feasible solution quickly, to then feed into a more sophisticated local search algorithm, which then tweaks the solution until a given stopping condition.  ‍

How route optimization algorithms work to solve the Travelling Salesman Problem.

Academic tsp solutions.

Academics have spent years trying to find the best solution to the Travelling Salesman Problem The following solutions were published in recent years:

• Machine learning speeds up vehicle routing : MIT researchers apply Machine Learning methods to solve large np-complete problems by solving sub-problems.
• Zero Suffix Method : Developed by Indian researchers, this method solves the classical symmetric TSP. 
• Biogeography‐based Optimization Algorithm : This method is designed based on the animals’ migration strategy to solve the problem of optimization. 
• Meta-Heuristic Multi Restart Iterated Local Search (MRSILS): The proponents of this research asserted that the meta-heuristic MRSILS is more efficient than the Genetic Algorithms when clusters are used. 
• Multi-Objective Evolutionary Algorithm : This method is designed for solving multiple TSP based on NSGA-II. 
• Multi-Agent System : This system is designed to solve the TSP of N cities with fixed resource. 

Real-world TSP applications

Despite the complexity of solving the Travelling Salesman Problem, it still finds applications in all verticals. 

For example, TSP solutions can help the logistics sector improve efficiency in the last mile. Last mile delivery is the final link in a supply chain, when goods move from a transportation hub, like a depot or a warehouse, to the end customer. Last mile delivery is also the leading cost driver in the supply chain. It costs an average of $10.1, but the customer only pays an average of $8.08 because companies absorb some of the cost to stay competitive. So bringing that cost down has a direct effect on business profitability.  

Minimizing costs in last mile delivery is essentially in last mile delivery is essentially a Vehicle Routing Problem (VRP). VRP, a generalized version of the travelling salesman problem, is one of the most widely studied problems in mathematical optimization. Instead of one best path, it deals with finding the most efficient set of routes or paths. The problem may involve multiple depots, hundreds of delivery locations, and several vehicles. As with the travelling salesman problem, determining the best solution to VRP is NP-complete.

Real-life TSP and VRP solvers

While academic solutions to TSP and VRP aim to provide the optimal solution to these NP-hard problems, many of them aren’t practical when solving real world problems, especially when it comes to solving last mile logistical challenges.

That’s because academic solvers strive for perfection and thus take a long time to compute the optimal solutions – hours, days, and sometimes years. If a delivery business needs to plan daily routes, they need a route solution within a matter of minutes. Their business depends on delivery route planning software so they can get their drivers and their goods out the door as soon as possible. Another popular alternative is to use Google maps route planner .

Real-life TSP and VRP solvers use route optimization algorithms that find a near-optimal solutions in a fraction of the time, giving delivery businesses the ability to plan routes quickly and efficiently.

If you want to know more about real-life TSP and VRP solvers, check out the resources below 👇

Route Optimization API - TSP Solver

Route Optimization API - VRP Solver

Frequently Asked Questions

Related articles.

Liked this article? See below for more recommended reading!

Solving the Vehicle Routing Problem (2024)

What Is Route Optimization?

How To Optimize Multi-Stop Routes With Google Maps (2024)

A business journal from the Wharton School of the University of Pennsylvania

Knowledge at Wharton Podcast

The ‘traveling salesman’ goes shopping: the efficiency of purchasing patterns in the grocery store, november 29, 2006 • 9 min listen.

If a traveling salesman has to visit a set number of cities in a set number of days, what is the shortest route he can take to cover all his stops and then return home?  And does the answer to this question relate in any way to the manner in which a shopper navigates her way through a grocery store? In a new paper, Wharton marketing professors Peter S. Fader and Eric T. Bradlow and doctoral student Sam K. Hui use a concept known as the "Traveling Salesman Problem" to study the efficiencies -- and inefficiencies -- of grocery shoppers.

More From Knowledge at Wharton

How ‘Thinking Out of the Shoebox’ Made Stuart Weitzman a Success

Chip Bergh, Former President & CEO of Levi Strauss & Co

Emily may, president, co-founder, and lead executive officer of right to be, looking for more insights.

Sign up to stay informed about our latest article releases.

• Coding Problems

Travelling Salesman Problem (TSP)

Problem statement, example 1: travelling salesman problem, example 2: travelling salesman problem, 1. simple approach, c++ code implementation, java code implementation, python code implementation, 2. travelling salesman problem using dynamic programming, c code implementation, 3. greedy approach, practice questions, frequently asked questions, 1. which algorithm is used for the travelling salesman problem, 2. what is the complexity of the travelling salesman problem, 3. how is this problem modelled as a graph problem, 4: what is the difficulty level of the travelling salesman problem.

Travelling Salesman Problem (TSP) – Given a set of cities and the distance between every pair of cities as an adjacency matrix, the problem is to find the shortest possible route that visits every city exactly once and returns to the starting point. The ultimate goal is to minimize the total distance travelled, forming a closed tour or circuit.

The TSP is referred to as an NP-hard problem, meaning there is no known algorithm to solve it in polynomial time for large instances. As the number of cities increases, the number of potential solutions grows exponentially, making an exhaustive search unfeasible. This complexity is one of the reasons why the TSP remains a popular topic of research. Learn More .

Input –

Confused about your next job?

Output –

Here, the TSP Tour is 0-2-1-3-0 and the cost of the tour is 48.

Minimum weight Hamiltonian Cycle : EACBDE= 32

Wondering how the Hamiltonian Cycle Problem and the Traveling Salesman Problem differ? The Hamiltonian Cycle problem is to find out if there exists a tour that visits each city exactly once. Here, we know that the Hamiltonian Tour exists (due to the graph being complete), and there are indeed many such tours. The problem is to find a minimum weight Hamiltonian Cycle.

There are various approaches to finding the solution to the travelling salesman problem- simple (naïve) approach, dynamic programming approach, and greedy approach. Let’s explore each approach in detail:

• Consider city 1 as the starting and ending point. Since the route is cyclic, we can consider any point as a starting point.
• Now, we will generate all possible permutations of cities which are (n-1)!.
• Find the cost of each permutation and keep track of the minimum cost permutation.
• Return the permutation with minimum cost.
• Time complexity: O(N!), Where N is the number of cities.
• Space complexity: O(1).

In the travelling salesman problem algorithm, we take a subset N of the required cities that need to be visited, the distance among the cities dist, and starting cities s as inputs. Each city is identified by a unique city id which we say like 1,2,3,4,5………n

Here we use a dynamic approach to calculate the cost function Cost(). Using recursive calls, we calculate the cost function for each subset of the original problem.

To calculate the cost(i) using Dynamic Programming , we need to have some recursive relation in terms of sub-problems.

We start with all subsets of size 2 and calculate C(S, i) for all subsets where S is the subset, then we calculate C(S, i) for all subsets S of size 3 and so on.

There are at most O(n2^n) subproblems, and each one takes linear time to solve. The total running time is, therefore, O(n^22^n). The time complexity is much less than O(n!) but still exponential. The space required is also exponential.

• Time Complexity: O(N^2*2^N).
• First of them is a list that can hold the indices of the cities in terms of the input matrix of distances between cities
• And the Second one is the array which is our result
• Perform traversal on the given adjacency matrix tsp[][] for all the cities and if the cost of reaching any city from the current city is less than the current cost update the cost.
• Generate the minimum path cycle using the above step and return their minimum cost.
• Time complexity: O(N^2*logN), Where N is the number of cities.
• Space complexity: O(N).
• City Tour Problem
• Shortest Common Substring

Ans . Travelling Salesman Problem uses Dynamic programming with a masking algorithm.

Ans.: The complexity of TSP using Greedy will be O(N^2 LogN) and using DP will be O(N^2 2^N).

Ans .: The TSP can be modelled as a graph problem by considering a complete graph G = (V, E). A tour is then a circuit in G that meets every node. In this context, tours are sometimes called Hamiltonian circuits.

Ans.: It is an NP-hard problem.

• Travelling Salesman Problem

Previous Post

Longest common subsequence, minimum spanning tree – kruskal algorithm.

Forgot password? New user? Sign up

Existing user? Log in

Traveling Salesperson Problem

Already have an account? Log in here.

A salesperson needs to visit a set of cities to sell their goods. They know how many cities they need to go to and the distances between each city. In what order should the salesperson visit each city exactly once so that they minimize their travel time and so that they end their journey in their city of origin?

The traveling salesperson problem is an extremely old problem in computer science that is an extension of the Hamiltonian Circuit Problem . It has important implications in complexity theory and the P versus NP problem because it is an NP-Complete problem . This means that a solution to this problem cannot be found in polynomial time (it takes superpolynomial time to compute an answer). In other words, as the number of vertices increases linearly, the computation time to solve the problem increases exponentially.

The following image is a simple example of a network of cities connected by edges of a specific distance. The origin city is also marked.

Network of cities

Here is the solution for that network, it has a distance traveled of only 14. Any other path that the salesman can takes will result in a path length that is more than 14.

Relationship to Graphs

Special kinds of tsp, importance for p vs np, applications.

The traveling salesperson problem can be modeled as a graph . Specifically, it is typical a directed, weighted graph. Each city acts as a vertex and each path between cities is an edge. Instead of distances, each edge has a weight associated with it. In this model, the goal of the traveling salesperson problem can be defined as finding a path that visits every vertex, returns to the original vertex, and minimizes total weight.

To that end, many graph algorithms can be used on this model. Search algorithms like breadth-first search (BFS) , depth-first search (DFS) , and Dijkstra's shortest path algorithm can certainly be used, however, they do not take into consideration that fact that every vertex must be visited.

The Traveling Salesperson Problem (TSP), an NP-Complete problem, is notoriously complicated to solve. That is because the greedy approach is so computational intensive. The greedy approach to solving this problem would be to try every single possible path and see which one is the fastest. Try this conceptual question to see if you have a grasp for how hard it is to solve.

For a fully connected map with \(n\) cities, how many total paths are possible for the traveling salesperson? Show Answer There are (n-1)! total paths the salesperson can take. The computation needed to solve this problem in this way grows far too quickly to be a reasonable solution. If this map has only 5 cities, there are \(4!\), or 24, paths. However, if the size of this map is increased to 20 cities, there will be \(1.22 \cdot 10^{17}\) paths!

The greedy approach to TSP would go like this:

• Find all possible paths.
• Find the cost of every paths.
• Choose the path with the lowest cost.

Another version of a greedy approach might be: At every step in the algorithm, choose the best possible path. This version might go a little quicker, but it's not guaranteed to find the best answer, or an answer at all since it might hit a dead end.

For NP-Hard problems (a subset of NP-Complete problems) like TSP, exact solutions can only be implemented in a reasonable amount of time for small input sizes (maps with few cities). Otherwise, the best approach we can do is provide a heuristic to help the problem move forward in an optimal way. However, these approaches cannot be proven to be optimal because they always have some sort of downside.

Small input sizes

As described, in a previous section , the greedy approach to this problem has a complexity of \(O(n!)\). However, there are some approaches that decrease this computation time.

The Held-Karp Algorithm is one of the earliest applications of dynamic programming . Its complexity is much lower than the greedy approach at \(O(n^2 2^n)\). Basically what this algorithm says is that every sub path along an optimal path is itself an optimal path. So, computing an optimal path is the same as computing many smaller subpaths and adding them together.

Heuristics are a way of ranking possible next steps in an algorithm in the hopes of cutting down computation time for the entire algorithm. They are often a tradeoff of some attribute - such as completeness, accuracy, or precision - in favor of speed. Heuristics exist for the traveling salesperson problem as well.

The most simple heuristic for this problem is the greedy heuristic. This heuristic simply says, at each step of the network traversal, choose the best next step. In other words, always choose the closest city that you have not yet visited. This heuristic seems like a good one because it is simple and intuitive, and it is even used in practice sometimes, however there are heuristics that are proven to be more effective.

Christofides algorithm is another heuristic. It produces at most 1.5 times the optimal weight for TSP. This algorithm involves finding a minimum spanning tree for the network. Next, it creates matchings for the cities of an odd degree (meaning they have an odd number of edges coming out of them), calculates an eulerian path , and converts back to a TSP path.

Even though it is typically impossible to optimally solve TSP problems, there are cases of TSP problems that can be solved if certain conditions hold.

The metric-TSP is an instance of TSP that satisfies this condition: The distance from city A to city B is less than or equal to the distance from city A to city C plus the distance from city C to city B. Or,

\[distance_{AB} \leq distance_{AC} + distance_{CB}\]

This is a condition that holds in the real world, but it can't always be expected to hold for every TSP problem. But, with this inequality in place, the approximated path will be no more than twice the optimal path. Even better, we can bound the solution to a \(3/2\) approximation by using Christofide's Algorithm .

The euclidean-TSP has an even stricter constraint on the TSP input. It states that all cities' edges in the network must obey euclidean distances . Recent advances have shown that approximation algorithms using euclidean minimum spanning trees have reduced the runtime of euclidean-TSP, even though they are also NP-hard. In practice, though, simpler heuristics are still used.

The P versus NP problem is one of the leading questions in modern computer science. It asks whether or not every problem whose solution can be verified in polynomial time by a computer can also be solved in polynomial time by a computer. TSP, for example, cannot be solved in polynomial time (at least that's what is currently theorized). However, TSP can be solved in polynomial time when it is phrased like this: Given a graph and an integer, x, decide if there is a path of length x or less than x . It's easy to see that given a proposed answer to this question, it is simple to check if it is less than or equal to x.

The traveling salesperson problem, like other problems that are NP-Complete, are very important to this debate. That is because if a polynomial time solution can be found to this problems, then \(P = NP\). As it stands, most scientists believe that \(P \ne NP\).

The traveling salesperson problem has many applications. The obvious ones are in the transportation space. Planning delivery routes or flight patterns, for example, would benefit immensly from breakthroughs is this problem or in the P versus NP problem .

However, this same logic can be applied to many facets of planning as well. In robotics, for instance, planning the order in which to drill holes in a circuit board is a complex task due to the sheer number of holes that must be drawn.

The best and most important application of TSP, however, comes from the fact that it is an NP-Complete problem. That means that its practical applications amount to the applications of any problem that is NP-Complete. So, if there are significant breakthroughs for TSP, that means that those exact same breakthrough can be applied to any problem in the NP-Complete class.

Problem Loading...

Note Loading...

Set Loading...

DSA Tutorial

Linked lists, stacks & queues, hash tables, shortest path, minimum spanning tree, maximum flow, time complexity, dsa reference, dsa examples, dsa the traveling salesman problem.

The Traveling Salesman Problem

The Traveling Salesman Problem states that you are a salesperson and you must visit a number of cities or towns.

Rules : Visit every city only once, then return back to the city you started in.

Goal : Find the shortest possible route.

Except for the Held-Karp algorithm (which is quite advanced and time consuming, (\(O(2^n n^2)\)), and will not be described here), there is no other way to find the shortest route than to check all possible routes.

This means that the time complexity for solving this problem is \(O(n!)\), which means 720 routes needs to be checked for 6 cities, 40,320 routes must be checked for 8 cities, and if you have 10 cities to visit, more than 3.6 million routes must be checked!

Note: "!", or "factorial", is a mathematical operation used in combinatorics to find out how many possible ways something can be done. If there are 4 cities, each city is connected to every other city, and we must visit every city exactly once, there are \(4!= 4 \cdot 3 \cdot 2 \cdot 1 = 24\) different routes we can take to visit those cities.

The Traveling Salesman Problem (TSP) is a problem that is interesting to study because it is very practical, but so time consuming to solve, that it becomes nearly impossible to find the shortest route, even in a graph with just 20-30 vertices.

If we had an effective algorithm for solving The Traveling Salesman Problem, the consequences would be very big in many sectors, like for example chip design, vehicle routing, telecommunications, and urban planning.

Checking All Routes to Solve The Traveling Salesman Problem

To find the optimal solution to The Traveling Salesman Problem, we will check all possible routes, and every time we find a shorter route, we will store it, so that in the end we will have the shortest route.

Good: Finds the overall shortest route.

Bad: Requires an awful lot of calculation, especially for a large amount of cities, which means it is very time consuming.

How it works:

• Check the length of every possible route, one route at a time.
• Is the current route shorter than the shortest route found so far? If so, store the new shortest route.
• After checking all routes, the stored route is the shortest one.

Such a way of finding the solution to a problem is called brute force .

Brute force is not really an algorithm, it just means finding the solution by checking all possibilities, usually because of a lack of a better way to do it.

Speed: {{ inpVal }}

Finding the shortest route in The Traveling Salesman Problem by checking all routes (brute force).

Progress: {{progress}}%

n = {{vertices}} cities

{{vertices}}!={{posRoutes}} possible routes

Show every route: {{showCompares}}

The reason why the brute force approach of finding the shortest route (as shown above) is so time consuming is that we are checking all routes, and the number of possible routes increases really fast when the number of cities increases.

Finding the optimal solution to the Traveling Salesman Problem by checking all possible routes (brute force):

Using A Greedy Algorithm to Solve The Traveling Salesman Problem

Since checking every possible route to solve the Traveling Salesman Problem (like we did above) is so incredibly time consuming, we can instead find a short route by just going to the nearest unvisited city in each step, which is much faster.

Good: Finds a solution to the Traveling Salesman Problem much faster than by checking all routes.

Bad: Does not find the overall shortest route, it just finds a route that is much shorter than an average random route.

• Visit every city.
• The next city to visit is always the nearest of the unvisited cities from the city you are currently in.
• After visiting all cities, go back to the city you started in.

This way of finding an approximation to the shortest route in the Traveling Salesman Problem, by just going to the nearest unvisited city in each step, is called a greedy algorithm .

Finding an approximation to the shortest route in The Traveling Salesman Problem by always going to the nearest unvisited neighbor (greedy algorithm).

As you can see by running this simulation a few times, the routes that are found are not completely unreasonable. Except for a few times when the lines cross perhaps, especially towards the end of the algorithm, the resulting route is a lot shorter than we would get by choosing the next city at random.

Finding a near-optimal solution to the Traveling Salesman Problem using the nearest-neighbor algorithm (greedy):

Other Algorithms That Find Near-Optimal Solutions to The Traveling Salesman Problem

In addition to using a greedy algorithm to solve the Traveling Salesman Problem, there are also other algorithms that can find approximations to the shortest route.

These algorithms are popular because they are much more effective than to actually check all possible solutions, but as with the greedy algorithm above, they do not find the overall shortest route.

Algorithms used to find a near-optimal solution to the Traveling Salesman Problem include:

• 2-opt Heuristic: An algorithm that improves the solution step-by-step, in each step removing two edges and reconnecting the two paths in a different way to reduce the total path length.
• Genetic Algorithm: This is a type of algorithm inspired by the process of natural selection and use techniques such as selection, mutation, and crossover to evolve solutions to problems, including the TSP.
• Simulated Annealing: This method is inspired by the process of annealing in metallurgy. It involves heating and then slowly cooling a material to decrease defects. In the context of TSP, it's used to find a near-optimal solution by exploring the solution space in a way that allows for occasional moves to worse solutions, which helps to avoid getting stuck in local minima.
• Ant Colony Optimization: This algorithm is inspired by the behavior of ants in finding paths from the colony to food sources. It's a more complex probabilistic technique for solving computational problems which can be mapped to finding good paths through graphs.

Time Complexity for Solving The Traveling Salesman Problem

To get a near-optimal solution fast, we can use a greedy algorithm that just goes to the nearest unvisited city in each step, like in the second simulation on this page.

Solving The Traveling Salesman Problem in a greedy way like that, means that at each step, the distances from the current city to all other unvisited cities are compared, and that gives us a time complexity of \(O(n^2) \).

But finding the shortest route of them all requires a lot more operations, and the time complexity for that is \(O(n!)\), like mentioned earlier, which means that for 4 cities, there are 4! possible routes, which is the same as \(4 \cdot 3 \cdot 2 \cdot 1 = 24\). And for just 12 cities for example, there are \(12! = 12 \cdot 11 \cdot 10 \cdot \; ... \; \cdot 2 \cdot 1 = 479,001,600\) possible routes!

See the time complexity for the greedy algorithm \(O(n^2)\), versus the time complexity for finding the shortest route by comparing all routes \(O(n!)\), in the image below.

But there are two things we can do to reduce the number of routes we need to check.

In the Traveling Salesman Problem, the route starts and ends in the same place, which makes a cycle. This means that the length of the shortest route will be the same no matter which city we start in. That is why we have chosen a fixed city to start in for the simulation above, and that reduces the number of possible routes from \(n!\) to \((n-1)!\).

Also, because these routes go in cycles, a route has the same distance if we go in one direction or the other, so we actually just need to check the distance of half of the routes, because the other half will just be the same routes in the opposite direction, so the number of routes we need to check is actually \( \frac{(n-1)!}{2}\).

But even if we can reduce the number of routes we need to check to \( \frac{(n-1)!}{2}\), the time complexity is still \( O(n!)\), because for very big \(n\), reducing \(n\) by one and dividing by 2 does not make a significant change in how the time complexity grows when \(n\) is increased.

To better understand how time complexity works, go to this page .

Actual Traveling Salesman Problems Are More Complex

The edge weight in a graph in this context of The Traveling Salesman Problem tells us how hard it is to go from one point to another, and it is the total edge weight of a route we want to minimize.

So far on this page, the edge weight has been the distance in a straight line between two points. And that makes it much easier to explain the Traveling Salesman Problem, and to display it.

But in the real world there are many other things that affects the edge weight:

• Obstacles: When moving from one place to another, we normally try to avoid obstacles like trees, rivers, houses for example. This means it is longer and takes more time to go from A to B, and the edge weight value needs to be increased to factor that in, because it is not a straight line anymore.
• Transportation Networks: We usually follow a road or use public transport systems when traveling, and that also affects how hard it is to go (or send a package) from one place to another.
• Traffic Conditions: Travel congestion also affects the travel time, so that should also be reflected in the edge weight value.
• Legal and Political Boundaries: Crossing border for example, might make one route harder to choose than another, which means the shortest straight line route might be slower, or more costly.
• Economic Factors: Using fuel, using the time of employees, maintaining vehicles, all these things cost money and should also be factored into the edge weights.

As you can see, just using the straight line distances as the edge weights, might be too simple compared to the real problem. And solving the Traveling Salesman Problem for such a simplified problem model would probably give us a solution that is not optimal in a practical sense.

It is not easy to visualize a Traveling Salesman Problem when the edge length is not just the straight line distance between two points anymore, but the computer handles that very well.

COLOR PICKER

Report Error

If you want to report an error, or if you want to make a suggestion, do not hesitate to send us an e-mail:

[email protected]

Top Tutorials

Top references, top examples, get certified.

The Network Effect

Beyond Supply Chains

Traveling Salesmen, Algorithms, and Supply Chain Complexity

More than just transportation, these problems are found throughout the supply chain. I liked this part:

Modeling the real world, with constraints like melting ice cream and idiosyncratic human behavior, is often where the real challenge lies. As mathematicians, operations research specialists, and corporate executives set out to mathematize and optimize the transportation networks that interconnect our modern world, they are re-discovering some of our most human quirks and capabilities. They are finding that their job is as much to discover the world, as it is to change it. The Traveling Salesman Problem, and its intellectual cousins, are far from theoretical; indeed, they are at the invisible heart of our transportation networks. Every time you want to go somewhere, or you want something to get to you, the chances are someone is thinking at that very moment how to make that process more efficient. We are all of us traveling salesmen.

Some of the best bits come from the author’s interviews with people at UPS:

But in trying to apply this mathematics to the real world of deliveries and drivers, UPS managers needed to learn that transportation is as much about people and the unique constraints they impose, as it is about negotiating intersections and time zones. As Jeff Winters put it to me, “on the surface, it should be very easy to come up with an optimized route and give it to the driver, and you’re done. We thought that would take a year.” That was a decade ago. For one thing, humans are irrational and prone to habit. When those habits are interrupted, interesting things happen. After the collapse of the I-35 bridge in Minnesota, for example, the number of travelers crossing the river, not surprisingly, dropped; but even after the bridge was restored, researcher David Levinson has noted, traffic levels never got near their previous levels again. People are also emotional, and it turns out an unhappy truck driver can be trouble. Modern routing models incorporate whether a truck driver is happy or not—something he may not know about himself. For example, one major trucking company that declined to be named does “predictive analysis” on when drivers are at greater risk of being involved in a crash. Not only does the company have information on how the truck is being driven—speeding, hard-braking events, rapid lane changes—but on the life of the driver.

If there’s a main takeaway from this piece, it’s that in the future, supply chain technology systems will be smarter, faster, and far more capable of handling complexity. Those who are making technology investments should plan for this new reality, and make sure their systems are designed to accommodate adaptation and change . I think it’s safe to say these systems of the future will not be ones that were designed twenty years ago–they will be an entirely different model .

Read the full article at Nautilus here.

Photo by Mark Harding

• Recent Posts

• The 4PL Revolution Roundup: Top 8 Opportunities for Today’s Logistics Providers - December 15, 2015
• Demand Sensing Round-Up (Blog Posts, White Papers, and Webinars!) - December 15, 2015
• Your Favorite Posts of 2015 - December 4, 2015

• Data Structures
• Linked List
• Binary Tree
• Binary Search Tree
• Segment Tree
• Disjoint Set Union
• Fenwick Tree
• Red-Black Tree
• Advanced Data Structures
• Dynamic Programming
• What is memoization? A Complete tutorial
• Dynamic Programming (DP) Tutorial with Problems
• Tabulation vs Memoization
• Optimal Substructure Property in Dynamic Programming | DP-2
• Overlapping Subproblems Property in Dynamic Programming | DP-1
• Steps for how to solve a Dynamic Programming Problem

Advanced Topics

• Bitmasking and Dynamic Programming | Set 1 (Count ways to assign unique cap to every person)
• Digit DP | Introduction
• Sum over Subsets | Dynamic Programming

Easy problems in Dynamic programming

• Count all combinations of coins to make a given value sum (Coin Change II)
• Subset Sum Problem
• Introduction and Dynamic Programming solution to compute nCr%p
• Cutting a Rod | DP-13
• Painting Fence Algorithm
• Longest Common Subsequence (LCS)
• Longest Increasing Subsequence (LIS)
• Longest subsequence such that difference between adjacents is one
• Maximum size square sub-matrix with all 1s
• Min Cost Path | DP-6
• Longest Common Substring (Space optimized DP solution)
• Count ways to reach the nth stair using step 1, 2 or 3
• Count Unique Paths in matrix
• Unique paths in a Grid with Obstacles

Medium problems on Dynamic programming

• 0/1 Knapsack Problem
• Printing Items in 0/1 Knapsack
• Unbounded Knapsack (Repetition of items allowed)
• Egg Dropping Puzzle | DP-11
• Word Break Problem | DP-32
• Vertex Cover Problem (Dynamic Programming Solution for Tree)
• Tile Stacking Problem
• Box Stacking Problem | DP-22
• Partition problem | DP-18

Travelling Salesman Problem using Dynamic Programming

• Longest Palindromic Subsequence (LPS)
• Longest Common Increasing Subsequence (LCS + LIS)
• Find all distinct subset (or subsequence) sums of an array
• Weighted Job Scheduling
• Count Derangements (Permutation such that no element appears in its original position)
• Minimum insertions to form a palindrome | DP-28
• Ways to arrange Balls such that adjacent balls are of different types

Hard problems on Dynamic programming

• Palindrome Partitioning
• Word Wrap Problem
• The Painter's Partition Problem
• Program for Bridge and Torch problem
• Matrix Chain Multiplication | DP-8
• Printing brackets in Matrix Chain Multiplication Problem
• Maximum sum rectangle in a 2D matrix | DP-27
• Maximum profit by buying and selling a share at most k times
• Minimum cost to sort strings using reversal operations of different costs
• Count of AP (Arithmetic Progression) Subsequences in an array
• Introduction to Dynamic Programming on Trees
• Maximum height of Tree when any Node can be considered as Root
• Longest repeating and non-overlapping substring
• Top 20 Dynamic Programming Interview Questions

Travelling Salesman Problem (TSP):  

Given a set of cities and the distance between every pair of cities, the problem is to find the shortest possible route that visits every city exactly once and returns to the starting point. Note the difference between Hamiltonian Cycle and TSP. The Hamiltonian cycle problem is to find if there exists a tour that visits every city exactly once. Here we know that Hamiltonian Tour exists (because the graph is complete) and in fact, many such tours exist, the problem is to find a minimum weight Hamiltonian Cycle. 

For example, consider the graph shown in the figure on the right side. A TSP tour in the graph is 1-2-4-3-1. The cost of the tour is 10+25+30+15 which is 80. The problem is a famous NP-hard problem. There is no polynomial-time know solution for this problem. The following are different solutions for the traveling salesman problem. 

Naive Solution:  

1) Consider city 1 as the starting and ending point.

2) Generate all (n-1)! Permutations of cities. 

3) Calculate the cost of every permutation and keep track of the minimum cost permutation. 

4) Return the permutation with minimum cost. 

Time Complexity: ?(n!) 

Dynamic Programming:  

Let the given set of vertices be {1, 2, 3, 4,….n}. Let us consider 1 as starting and ending point of output. For every other vertex I (other than 1), we find the minimum cost path with 1 as the starting point, I as the ending point, and all vertices appearing exactly once. Let the cost of this path cost (i), and the cost of the corresponding Cycle would cost (i) + dist(i, 1) where dist(i, 1) is the distance from I to 1. Finally, we return the minimum of all [cost(i) + dist(i, 1)] values. This looks simple so far. 

Now the question is how to get cost(i)? To calculate the cost(i) using Dynamic Programming, we need to have some recursive relation in terms of sub-problems. 

Let us define a term C(S, i) be the cost of the minimum cost path visiting each vertex in set S exactly once, starting at 1 and ending at i . We start with all subsets of size 2 and calculate C(S, i) for all subsets where S is the subset, then we calculate C(S, i) for all subsets S of size 3 and so on. Note that 1 must be present in every subset.

Below is the dynamic programming solution for the problem using top down recursive+memoized approach:-

For maintaining the subsets we can use the bitmasks to represent the remaining nodes in our subset. Since bits are faster to operate and there are only few nodes in graph, bitmasks is better to use.

For example: –  

10100 represents node 2 and node 4 are left in set to be processed

010010 represents node 1 and 4 are left in subset.

NOTE:- ignore the 0th bit since our graph is 1-based

Time Complexity : O(n 2 *2 n ) where O(n* 2 n) are maximum number of unique subproblems/states and O(n) for transition (through for loop as in code) in every states.

Auxiliary Space: O(n*2 n ), where n is number of Nodes/Cities here.

For a set of size n, we consider n-2 subsets each of size n-1 such that all subsets don’t have nth in them. Using the above recurrence relation, we can write a dynamic programming-based solution. There are at most O(n*2 n ) subproblems, and each one takes linear time to solve. The total running time is therefore O(n 2 *2 n ). The time complexity is much less than O(n!) but still exponential. The space required is also exponential. So this approach is also infeasible even for a slightly higher number of vertices. We will soon be discussing approximate algorithms for the traveling salesman problem.

Next Article: Traveling Salesman Problem | Set 2  

References:  

http://www.lsi.upc.edu/~mjserna/docencia/algofib/P07/dynprog.pdf  

http://www.cs.berkeley.edu/~vazirani/algorithms/chap6.pdf  

Please Login to comment...

Similar reads.

• Google Releases ‘Prompting Guide’ With Tips For Gemini In Workspace
• Google Cloud Next 24 | Gmail Voice Input, Gemini for Google Chat, Meet ‘Translate for me,’ & More
• 10 Best Viber Alternatives for Better Communication
• 12 Best Database Management Software in 2024
• 30 OOPs Interview Questions and Answers (2024)

Improve your Coding Skills with Practice

What kind of Experience do you want to share?

To revisit this article, visit My Profile, then View saved stories .

• Backchannel
• Newsletters
• WIRED Insider
• WIRED Consulting

Duncan Geere

Film explores moral and political repercussions of proving P equals NP

What would happen if one government learnt how to bypass the world's encryption systems? That's the all-too-timely question asked by Travelling Salesman , a movie that hands the starring role to a mathematical puzzle.

Its title refers to the "travelling salesman problem" -- given a list of cities and the distances between them all, what's the most efficient way that a salesman can visit them all and return home?

It's easy to check whether a given answer satisfies the criteria, but finding the best solution without trying every single possibility may be impossible. This puzzle, and others like it, are referred to as P vs NP.

In Travelling Salesman , a quartet of mathematicians solve this problem -- they find a way to shortcut the brute-force process of trying every approach, instantaneously negating the way the world's encryption systems operates. They've been hired to do their work by the United States, but they're confronted with the moral quandary of whether to hand over their research to the government or publish it publicly for the whole world's benefit. "We wanted to make a film that felt authentic to a niche group, but also ultimately would be accessible to a wide audience," said Tim Lanzone, one of the movie's creators. "The film is science fiction, but we wanted the science part to be as accurate as possible," his brother and co-writer, Andy Lanzone, added.

It's set almost entirely around a conference table in a sleek, modern, government facility. Initially the four argue over the implications of their research, before a representative of the government arrives and their argument escalates significantly. "I wanted to see if I could contain drama in one room," says Tim Lanzone. "Film school professors always warn that the hardest and most difficult scenes to shoot occur around kitchen tables."

Of course, that makes shooting it cheap, too. The film has a definite DIY aesthetic, but it's impressive what Lanzone has managed to achieve within those constraints. He opted deliberately for a cool colour temperature to make the atmosphere intense and unsettling. "Most of the drama takes place in one room, so we looked to use any sort of additional drama we could harness to our advantage."

That atmosphere is paired with fast-paced and complex dialogue -- it's not a movie that you can put on in the background. Attempts are made to explain the mathematical ideas behind the plot, but you will get more out of it if you've done a little research beforehand. The Lanzones admit this. "Understanding P vs NP directly isn't essential to enjoying the film -- especially if you have some grasp of the implications," says Tim. "We try to simplify it as well as we can during certain scenes but the pace of the dialogue is so fast that a lot of people are just along for the ride."

The movie isn't about the maths, though. Travelling Salesman is a film about power, and how it should be distributed. The government representative's job is to convince the four to sign off their parts of the solution, but it's an uphill struggle as they grapple with the repercussions of what the knowledge would mean for the world.

The bogeyman of China is brought up repeatedly, but it's fascinating to watch the film in light of recent revelations about the extent to which the US and UK's security agencies are already violating global encryption systems through back doors created in many of the world's communication networks.

David Gilbert

Matt Burgess

Lauren Goode

The script was written long before Edward Snowden's revelations came to light, but Tim Lanzone says the topic had been on his mind for some time. "I wrote the first draft of the script back in early 2009, which in cyber-time is like an eternity. Back then, articles were really starting to come out more and more highlighting issues related to cyber-security and cyber-espionage -- particularly on important defence targets. It just seemed to be the next cold war-style battleground."

Andy Lanzone adds: "I don't know anyone that foresaw the scope of what the NSA is doing, so that was entirely coincidental. But the more general topic of cyber warfare was an interesting topic that we felt would only become more relevant over time."

The film firmly fails the Bechdel Test in that every one of the characters is white and male, limiting their perspectives considerably. The Lanzones blame this lack of diversity on budget: "We shot the film for under $25,000 and had a whirlwind casting session over two days in LA -- which was all the time we could afford," says Tim. "Knowing we only had about ten days to shoot Travelling Salesman , we were just trying to narrow it down to the five people who could most ably handle the complex language."

Ultimately, you'll get the same amount out of Travelling Salesman that you put into it. It's not a casual watch, but rewards concentration and occasionally pausing the movie and opening Wikipedia to check whether you've understood something right. If you were being cruel, you'd call it an argument for an hour and a half. But the themes and issues it addresses have never been more relevant -- the power of access to private communications, and whether it's right for that power to be concentrated under the auspices of one or a few governments. If you have strong feelings on the matter, on either side, Travelling Salesman is an essential watch.

This article was originally published by WIRED UK

Marah Eakin

Angela Watercutter

Jennifer M. Wood

Eric Ravenscraft

Megan Farokhmanesh

• Share full article

Planning to Combine Business and Leisure Travel? You’re Not Alone.

As employees increasingly add leisure time to their business trips, companies are trying to figure out where their duty of care obligations begin and end.

Credit... Aart-Jan Venema

Supported by

By Amy Zipkin

• April 7, 2024

On a Sunday in late January, Melinda Buchmann, who lives in Florida and supervises client relations for RevShoppe, a 30-person remote company advising organizations on sales techniques and strategies, arrived in Banff, Alberta, to help set up a four-day company meeting.

The last day of the event, her husband, Josh, a director of strategic partnerships for the delivery company DoorDash , who also works remotely, joined her. They spent two leisurely days hiking in Banff National Park and visiting Lake Louise.

“I take advantage, because I don’t know when I’m going to return,” Ms. Buchmann said of the decision to combine downtime with a business trip.

As postpandemic work life has changed, and arrangements now include full-time office attendance as well as hybrid and remote work, so, too, has business travel. The phenomenon known as bleisure, or blended business and leisure travel, was initially embraced largely by digital nomads . But such combined travel is now also popular with people outside that group . Allied Market Research, a subsidiary of Allied Analytics, based in Portland, Ore., estimated that the bleisure travel market was $315.3 billion in 2022 and would reach $731.4 billion by 2032.

As employees increasingly add leisure time to their business trips, companies are struggling to determine where their legal obligation to protect employees from harm — their so-called duty of care — begins and ends. And workers may think that because their trip started with business, they will get all the help they need if something goes wrong on the leisure end. Instead, they should generally consider the leisure part of a trip as a regular vacation where they cover all expenses and contingencies.

Companies are responsible for knowing where their employees are during a business trip, covering expenses if an accident or emergency occurs, securing new lodging if a hotel is damaged, even swapping out a broken down rental car. Still, it’s not entirely clear if that coverage ends completely after the conference or the last client meeting.

Companies recognize that threats are increasing, said Robert Cole, senior research analyst focusing on lodging and leisure travel at Phocuswright, a market research company. They are trying to figure out how to take care of a valuable company resource, the employee, without leaving themselves open to financial risk or potential litigation.

“Crafting a comprehensive policy that balances business objectives, employee well-being and legal considerations can be challenging,” Nikolaos Gkolfinopoulos, head of tourism at ICF, a consulting and technology services company in Reston, Va., wrote in an email.

Employees may be on their own without realizing it and may be surprised by out-of-pocket expenses if they require hospital care abroad or evacuation, said Suzanne Morrow, chief executive of InsureMyTrip , an online insurance travel comparison site in Warwick, R.I.

Ms. Morrow said medical coverage provided by a company “is generally only for the dates of the actual business trip abroad.” If travelers are extending the trip for personal travel, she added, “they would want to secure emergency medical coverage for that additional time abroad.”

Employers and employees are left to figure out when the business portion of the trip ends and the leisure segment begins, a significant detail if an employee has a medical emergency. “Where does the corporation liability end?” said Kathy Bedell, senior vice president at BCD Travel, a travel management company.

Companies have varying policies to deal with the new travel amalgam. The chief executive of RevShoppe, Patricia McLaren, based in Austin, Texas, said the company provided flexible travel options and allowed employees to work anywhere they choose.

Even so, there are constraints. The company requires all employees, including executives, to sign liability and insurance waivers when they are on a voluntary company-sponsored trip, such as an off-site meeting. Such waivers typically place responsibility on employees for their own well-being. And if they bring someone, they are responsible for that person’s expenses.

Employees are responsible for requesting the paid time off and notifying their managers of their whereabouts, although that part is not a requirement. Managers have to ensure adequate staffing, Ms. McLaren said.

Elsewhere, employees may not bother to mention the leisure portion of their trip. Eliot Lees, a vice president and managing director at ICF, said he had been on trips as a child with his parents when they combined business and leisure. His parents were academics, who would piggyback vacations onto conferences.

Now he does the same. “I don’t think I ever asked for approval,” he said. (ICF has no formal business-leisure travel policy. It’s allowed as part of personal time off.) After a conference in the Netherlands last year, he spent four days hiking in the northern part of the country.

“I go anywhere, and take more risks than I should,” he said. He said he didn’t carry personal travel or accident insurance.

Any nonchalance may quickly evaporate if a threat emerges. Security experts say even low-risk locations can become high-risk for a few days or weeks of the year.

“Companies are concerned about losing visibility into a traveler’s whereabouts if they booked flights and hotels outside their corporate travel management company,” Benjamin Thorne, senior intelligence manager in London for Crisis24, a subsidiary of GardaWorld, wrote in an email. “The company may think the traveler is in one city when, in reality, they could have booked a holiday package to another nearby city. This lack of visibility by the company makes it difficult to support travelers when a disaster occurs.”

He also raised the possibility that “a traveler with bleisure travel reservations and expectations may find their work trip canceled due to changes in the risk environment or company policy, disrupting their leisure plans.”

Will a company step in off hours if there’s a problem? “That depends on how you are booked,” Mr. Cole, the senior research analyst at Phocuswright, said. A rule of thumb is the further you get from corporate control, the greater the gray area gets.

Half of GoldSpring Consulting’s clients take the responsibility for the entire trip, said Will Tate, a partner at the consultancy based in Cross Roads, Texas, and a certified public accountant. They don’t want the reputational risk. The other half say: “The business trip ended Friday. That’s when we end our duty of care.”

Some companies are trying to define and narrow the gray area. “If you are clearly on personal time, there is no legal requirement for your employer to provide for you,” said Nicole Page, a lawyer whose practice includes employment law at Reavis Page Jump in New York.

Uber provides employees with advisories before a trip, travel assessments, safety tips while traveling and emergency travel assistance, including medical aid, airport travel support, urgent and emergency assistance, and lost or stolen personal property insurance whether they are on business or pleasure travel or a combination.

And at DoorDash, Chris Cherry, head of global safety and security, wrote in an email that “while personal travel is not something we track, we have received requests to extend our travel support capabilities to personal travel.” Mr. Cherry said in those cases, the company has manually added employee leisure itineraries to its travel risk management system and “provided the same level of overwatch that we do for regular business travel.”

The Buchmanns plan to travel this month to Barcelona, Spain, for the McDonald’s Worldwide Convention. DoorDash will have a booth, and Mr. Buchmann will work on the exhibit floor and also entertain clients.

Ms. Buchmann will accompany him. She plans to go sightseeing in the morning, and work in the afternoons and evenings Barcelona time. She will also take three days of paid time off and has shared her plans with Ms. McLaren, the RevShoppe chief executive.

They will stay a day after the conference and plan to visit the Dalí Theater and Museum in Figueres. “I’m sure there will be no shortage of tapas and window shopping along way,” Mr. Buchmann said. He expects to be back at work the next Monday.

Explore Our Business Coverage

Dive deeper into the people, issues and trends shaping the world of business..

Stopping a Huge Cyberattack: A Microsoft engineer noticed something was off on a piece of software he worked on. He soon discovered someone was probably trying to gain access to computers all over the world .

Hoping for an A.I. Productivity Boost:  Economists doubt that A.I. is already visible in productivity data . Big companies, however, talk often about adopting it to improve efficiency.

Cashing In on Graffiti:  Brands, developers and even officials are embracing the global appeal of street art , but the boom comes with questions about preserving a neighborhood’s cultural cachet.

‘Twitter Menace’ or True Believer?: The deep-pocketed tech investor Garry Tan says he wants to save San Francisco. But his pugnacious online habits are making him enemies .

A C.E.O.’s Bold Claims:  Amira Yahyaoui, a human rights activist, promoted the success of her student aid start-up, Mos. Some of her statements do not add up .

Advertisement

Turn Your Curiosity Into Discovery

Latest facts.

Follistatin344 Peptide Considerations

Approach for Using 5 Tips To Help You Write Your Dissertation

40 facts about elektrostal.

Written by Lanette Mayes

Modified & Updated: 02 Mar 2024

Reviewed by Jessica Corbett

Elektrostal is a vibrant city located in the Moscow Oblast region of Russia. With a rich history, stunning architecture, and a thriving community, Elektrostal is a city that has much to offer. Whether you are a history buff, nature enthusiast, or simply curious about different cultures, Elektrostal is sure to captivate you.

This article will provide you with 40 fascinating facts about Elektrostal, giving you a better understanding of why this city is worth exploring. From its origins as an industrial hub to its modern-day charm, we will delve into the various aspects that make Elektrostal a unique and must-visit destination.

So, join us as we uncover the hidden treasures of Elektrostal and discover what makes this city a true gem in the heart of Russia.

Key Takeaways:

• Elektrostal, known as the “Motor City of Russia,” is a vibrant and growing city with a rich industrial history, offering diverse cultural experiences and a strong commitment to environmental sustainability.
• With its convenient location near Moscow, Elektrostal provides a picturesque landscape, vibrant nightlife, and a range of recreational activities, making it an ideal destination for residents and visitors alike.

Known as the “Motor City of Russia.”

Elektrostal, a city located in the Moscow Oblast region of Russia, earned the nickname “Motor City” due to its significant involvement in the automotive industry.

Home to the Elektrostal Metallurgical Plant.

Elektrostal is renowned for its metallurgical plant, which has been producing high-quality steel and alloys since its establishment in 1916.

Boasts a rich industrial heritage.

Elektrostal has a long history of industrial development, contributing to the growth and progress of the region.

Founded in 1916.

The city of Elektrostal was founded in 1916 as a result of the construction of the Elektrostal Metallurgical Plant.

Located approximately 50 kilometers east of Moscow.

Elektrostal is situated in close proximity to the Russian capital, making it easily accessible for both residents and visitors.

Known for its vibrant cultural scene.

Elektrostal is home to several cultural institutions, including museums, theaters, and art galleries that showcase the city’s rich artistic heritage.

A popular destination for nature lovers.

Surrounded by picturesque landscapes and forests, Elektrostal offers ample opportunities for outdoor activities such as hiking, camping, and birdwatching.

Hosts the annual Elektrostal City Day celebrations.

Every year, Elektrostal organizes festive events and activities to celebrate its founding, bringing together residents and visitors in a spirit of unity and joy.

Has a population of approximately 160,000 people.

Elektrostal is home to a diverse and vibrant community of around 160,000 residents, contributing to its dynamic atmosphere.

Boasts excellent education facilities.

The city is known for its well-established educational institutions, providing quality education to students of all ages.

A center for scientific research and innovation.

Elektrostal serves as an important hub for scientific research, particularly in the fields of metallurgy, materials science, and engineering.

Surrounded by picturesque lakes.

The city is blessed with numerous beautiful lakes, offering scenic views and recreational opportunities for locals and visitors alike.

Well-connected transportation system.

Elektrostal benefits from an efficient transportation network, including highways, railways, and public transportation options, ensuring convenient travel within and beyond the city.

Famous for its traditional Russian cuisine.

Food enthusiasts can indulge in authentic Russian dishes at numerous restaurants and cafes scattered throughout Elektrostal.

Home to notable architectural landmarks.

Elektrostal boasts impressive architecture, including the Church of the Transfiguration of the Lord and the Elektrostal Palace of Culture.

Offers a wide range of recreational facilities.

Residents and visitors can enjoy various recreational activities, such as sports complexes, swimming pools, and fitness centers, enhancing the overall quality of life.

Provides a high standard of healthcare.

Elektrostal is equipped with modern medical facilities, ensuring residents have access to quality healthcare services.

Home to the Elektrostal History Museum.

The Elektrostal History Museum showcases the city’s fascinating past through exhibitions and displays.

A hub for sports enthusiasts.

Elektrostal is passionate about sports, with numerous stadiums, arenas, and sports clubs offering opportunities for athletes and spectators.

Celebrates diverse cultural festivals.

Throughout the year, Elektrostal hosts a variety of cultural festivals, celebrating different ethnicities, traditions, and art forms.

Electric power played a significant role in its early development.

Elektrostal owes its name and initial growth to the establishment of electric power stations and the utilization of electricity in the industrial sector.

Boasts a thriving economy.

The city’s strong industrial base, coupled with its strategic location near Moscow, has contributed to Elektrostal’s prosperous economic status.

Houses the Elektrostal Drama Theater.

The Elektrostal Drama Theater is a cultural centerpiece, attracting theater enthusiasts from far and wide.

Popular destination for winter sports.

Elektrostal’s proximity to ski resorts and winter sport facilities makes it a favorite destination for skiing, snowboarding, and other winter activities.

Promotes environmental sustainability.

Elektrostal prioritizes environmental protection and sustainability, implementing initiatives to reduce pollution and preserve natural resources.

Home to renowned educational institutions.

Elektrostal is known for its prestigious schools and universities, offering a wide range of academic programs to students.

Committed to cultural preservation.

The city values its cultural heritage and takes active steps to preserve and promote traditional customs, crafts, and arts.

Hosts an annual International Film Festival.

The Elektrostal International Film Festival attracts filmmakers and cinema enthusiasts from around the world, showcasing a diverse range of films.

Encourages entrepreneurship and innovation.

Elektrostal supports aspiring entrepreneurs and fosters a culture of innovation, providing opportunities for startups and business development.

Offers a range of housing options.

Elektrostal provides diverse housing options, including apartments, houses, and residential complexes, catering to different lifestyles and budgets.

Home to notable sports teams.

Elektrostal is proud of its sports legacy, with several successful sports teams competing at regional and national levels.

Boasts a vibrant nightlife scene.

Residents and visitors can enjoy a lively nightlife in Elektrostal, with numerous bars, clubs, and entertainment venues.

Promotes cultural exchange and international relations.

Elektrostal actively engages in international partnerships, cultural exchanges, and diplomatic collaborations to foster global connections.

Surrounded by beautiful nature reserves.

Nearby nature reserves, such as the Barybino Forest and Luchinskoye Lake, offer opportunities for nature enthusiasts to explore and appreciate the region’s biodiversity.

Commemorates historical events.

The city pays tribute to significant historical events through memorials, monuments, and exhibitions, ensuring the preservation of collective memory.

Promotes sports and youth development.

Elektrostal invests in sports infrastructure and programs to encourage youth participation, health, and physical fitness.

Hosts annual cultural and artistic festivals.

Throughout the year, Elektrostal celebrates its cultural diversity through festivals dedicated to music, dance, art, and theater.

Provides a picturesque landscape for photography enthusiasts.

The city’s scenic beauty, architectural landmarks, and natural surroundings make it a paradise for photographers.

Connects to Moscow via a direct train line.

The convenient train connection between Elektrostal and Moscow makes commuting between the two cities effortless.

A city with a bright future.

Elektrostal continues to grow and develop, aiming to become a model city in terms of infrastructure, sustainability, and quality of life for its residents.

In conclusion, Elektrostal is a fascinating city with a rich history and a vibrant present. From its origins as a center of steel production to its modern-day status as a hub for education and industry, Elektrostal has plenty to offer both residents and visitors. With its beautiful parks, cultural attractions, and proximity to Moscow, there is no shortage of things to see and do in this dynamic city. Whether you’re interested in exploring its historical landmarks, enjoying outdoor activities, or immersing yourself in the local culture, Elektrostal has something for everyone. So, next time you find yourself in the Moscow region, don’t miss the opportunity to discover the hidden gems of Elektrostal.

Q: What is the population of Elektrostal?

A: As of the latest data, the population of Elektrostal is approximately XXXX.

Q: How far is Elektrostal from Moscow?

A: Elektrostal is located approximately XX kilometers away from Moscow.

Q: Are there any famous landmarks in Elektrostal?

A: Yes, Elektrostal is home to several notable landmarks, including XXXX and XXXX.

Q: What industries are prominent in Elektrostal?

A: Elektrostal is known for its steel production industry and is also a center for engineering and manufacturing.

Q: Are there any universities or educational institutions in Elektrostal?

A: Yes, Elektrostal is home to XXXX University and several other educational institutions.

Q: What are some popular outdoor activities in Elektrostal?

A: Elektrostal offers several outdoor activities, such as hiking, cycling, and picnicking in its beautiful parks.

Q: Is Elektrostal well-connected in terms of transportation?

A: Yes, Elektrostal has good transportation links, including trains and buses, making it easily accessible from nearby cities.

Q: Are there any annual events or festivals in Elektrostal?

A: Yes, Elektrostal hosts various events and festivals throughout the year, including XXXX and XXXX.

Was this page helpful?

Our commitment to delivering trustworthy and engaging content is at the heart of what we do. Each fact on our site is contributed by real users like you, bringing a wealth of diverse insights and information. To ensure the highest standards of accuracy and reliability, our dedicated editors meticulously review each submission. This process guarantees that the facts we share are not only fascinating but also credible. Trust in our commitment to quality and authenticity as you explore and learn with us.

Share this Fact:

• Vacation Rentals
• Restaurants
• Things to do
• Elektrostal Tourism
• Elektrostal Hotels
• Elektrostal Bed and Breakfast
• Flights to Elektrostal
• Elektrostal Restaurants
• Things to Do in Elektrostal
• Elektrostal Travel Forum
• Elektrostal Photos
• Elektrostal Map
• All Elektrostal Hotels
• Elektrostal Hotel Deals
• Elektrostal Hostels
• Elektrostal Business Hotels
• Elektrostal Family Hotels
• Elektrostal Spa Resorts
• 3-stars Hotels in Elektrostal
• Elektrostal Hotels with Game room
• Elektrostal Hotels with Banquet hall
• Hotels near Electrostal History and Art Museum
• Hotels near Park of Culture and Leisure
• Hotels near Statue of Lenin
• Hotels near Museum and Exhibition Center
• Hotels near Museum of Labor Glory
• Hotels near (ZIA) Zhukovsky International Airport
• Hotels near (VKO) Vnukovo Airport
• Hotels near (DME) Domodedovo Airport
• Dreams Jade Resort & Spa
• Xafira Deluxe Resort & Spa
• Mandalay Bay Resort & Casino
• The LINQ Hotel + Experience
• Bellagio Las Vegas
• Hyatt Ziva Cap Cana
• Isla Bella Beach Resort
• Secrets Akumal Riviera Maya
• Hotel Xcaret Mexico
• Hotel Tapasoli
• Excellence Punta Cana
• Hilton Hawaiian Village Waikiki Beach Resort
• Hotel Chester
• Barcelo Bavaro Palace All Inclusive Resort
• Aruba Ocean Villas
• Popular All-Inclusive Resorts
• Popular Beach Resorts
• Popular Family Resorts
• Popular All-Inclusive Hotels
• Popular Hotels With Waterparks
• Popular Honeymoon Resorts
• Popular Luxury Resorts
• Popular All-Inclusive Family Resorts
• Popular Golf Resorts
• Popular Spa Resorts
• Popular Cheap Resorts
• All Elektrostal Restaurants
• Cafés in Elektrostal
• Chinese Restaurants in Elektrostal
• European Restaurants for Families in Elektrostal
• European Restaurants for Large Groups in Elektrostal
• European Restaurants for Lunch in Elektrostal
• Fast Food Restaurants in Elektrostal
• French Restaurants in Elektrostal
• Italian Restaurants in Elektrostal
• Japanese Restaurants in Elektrostal
• Pizza in Elektrostal
• Russian Restaurants in Elektrostal
• Seafood Restaurants in Elektrostal
• Vegetarian Restaurants in Elektrostal
• GreenLeaders
• Elektrostal
• Things to Do
• Travel Stories
• Rental Cars
• Add a Place
• Travel Forum
• Travelers' Choice
• Help Center
• Europe    
• Russia    
• Central Russia    
• Moscow Oblast    
• Elektrostal    
• Elektrostal Restaurants    

Ratings and reviews

Location and contact.

PEKIN, Elektrostal - Lenina Ave. 40/8 - Restaurant Reviews, Photos & Phone Number - Tripadvisor