# Essay Example: The Traveling Salesman Problem (TSP)

Published: 2023-03-06 Type of paper: Research paper Categories: Human resources Problem solving Sales Mathematics Pages: 7 Wordcount: 1655 words
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The traveling salesman problem is an algorithmic problem used to find the shortest way or route between two or more set points and locations that required to be visited. From the problem statement, the points stated are the cities that a salesperson might be in dire need to visit. The main objective of the salesman is to manage and keep both costs and distance being traveled to be very low. The traveling salesman applied computer science knowledge to get a reliable and sufficient root, which can enable data to move between different nodes. It requires the identification of a hardware optimization method to be used. Is your time best spent reading someone else’s essay? Get a 100% original essay FROM A CERTIFIED WRITER!

Two mathematicians first developed the application, thus W.R Hamilton from Irish and Thomas Kirkman from British. They created a game that was solvable by finding Hamilton's cycle. With research and study of TSP, which has taken place for decades, many solutions have been theorized (Reinelt, 1994). It is recommended that for the simplest solution, all possibilities must have existed even though this method and solution tend to be time consuming and most expensive. The use of heuristics in the solution provides probability outcomes. The results attained from the process are treated as approximate but not optimal. Other solutions that can be used include branch and bond, Monte Carlo, and the Las Vegas algorithms. The TSP concentrates more on finding the solution, which is cheapest rather than focusing on the practical route.

The challenge is created by a large number of variables in the process of finding the shortest route, which are approximate fast and cheap solutions. The traveling salesman can arise in various contexts. It can be applied in the field of computer wiring, vehicle routing, and even in job shop scheduling. An excellent example of TSP is when a given set of cities and distances between different pairs of cities, one will try to find the shortest way possible (Boese, 1995). The use of the Hamiltonian cycle provides the solution of the most concise and cheapest distance.

## Example

From the figure above, when the business person travels from 1-2-4-3-1, and the cost used to cover the distances is 10+25+30+15, which gives a total of 80.In such a problem, even though it is complicated, but different solutions can be applied. When applying dynamic programming. We start by naming the vertices 1, 2, 3, 4...n, and we start from 1 and getting different ending point output. And the solution will also assist us in getting the cost used to cover the distance.

Explanations of Three Algorithms Used To Obtain an Initial Solution for TSP: Random Algorithm, Iterative Random Algorithm, and Greedy Algorithm

## Random Algorithm

Traveling salesman problems requires on to find tour ways through different cities, and every city is visited once and only once, then returning to the starting point using the root, which is the shortest. There are different ways of determining and estimating the length of the distance of the salesman during the optimal tour, but this can be done through a set of random assignments. The random algorithm is where the distance covered by the salesperson is being determined by choosing the sample of tour randomly to be used from a different set of assignments. The approach uses the lengths from randomly generated trips.

From most experiences with the standard test problems, we find that even the shortest distance of 20,000 miles will be very different and far from ordinary. But will apply the random algorithm to find the length of the shortest distance covered.

## Iterative Random Forests

The iterative Random Forest algorithm is described as the most efficient approach which can be used to search for the interaction between unknown form and order where there is high dimensional data. The iterative random forests also help specifically in providing the means which can be used in interpreting fitted random forests. Also, it helps in identifying features combinations that are used in the decision making of the paths in the tree ensemble. The iterative random forest is being used as the tool for generating the hypothesis when finding the solution of different problems like in developmental biology and even in the precision of medicine.

## Greedy Algorithms

The greedy algorithm is a strategy whereby different sets of resources are being divided, basing the decision on the maximum and the immediate availability of the funds to be used. In solving the problems using the greedy approach, there are two different stages to be followed, thus: scanning the list of the items available and the optimization. The two steps are being covered parallel by using the division of array (Gomez et al., 2011). To clearly understand the greedy approach, one must have a working knowledge of context switching and recursion. The experience of context switching and recursion assist in understanding how to trace the code. The greedy paradigm can also be defined in the individual necessary and sufficient statements. The conditions for determining the greedy algorithms involve situations where each stepwise solution structure a problem and the best solution, which is accepted. And also, sufficient if the structuring of a given problem can change the number of greedy steps.

## How is the Greedy Strategy Used to Improve the Starting Solution Obtained In 2

The mercenary logic and strategy are defined by the previous approach, which was taken in advance in each algorithm stage (Ahuja, Orlin, & Tiwari, 2000). A good example is where the Djikstras algorithm applied the greedy logic to make the identification of different hosts on the internet and to determine and calculate the cost function, which is involved. The value which is returned by calculating the cost function will decide whether the path is greedy or non-greedy. It is logically right to state that the algorithm will cease to be greedy if the step taken at any given stage is not greedy.

There are several characteristics of the greedy approach strategy, and the most important features include the presence of an ordered list of resources and the cost attributions. On the greedy strategy, there are several quantify constraints on a system. When the greedy algorithm is applied, there is a need to take the maximum quantity, which is applied during the obstacle. There are a number of reasons and the purpose of using the greedy approach. The approach has few tradeoffs, which make it more suitable for optimization. Another good reason for using the greedy strategy is the most appropriate solution, in a condition where there are many activities, the decision must be made to finish the current business, but the activities can be managed and performed within the same time.

The greedy strategy can be applied to divide the problem based on the condition, taking into consideration not to combine the solutions. The recursive division, which is in the activity selection problem, is being achieved by scanning different items not more than once and putting into consideration certain activities.

The greedy algorithm structure takes all the available data, which is in a given problem, and after that, the rule for elements is set, which requires the solution of each step to be added as the process continues. On the graph application, all the most significant numbers will be se4lected at each level of the graph. The solution will be found by summing up all the choices which are found. The exact greedy algorithms which can be used to solve the problem include the greedy choice property, and this is where the optimal solution is attained by choosing the available optimal choice at each given step. Another strategy is the optimal substructure, and this is where the whole problem contains the optimal solution for the subproblems.

It is crucial to note that greedy algorithms always work on the problems, and it is right to note that every step taken is very vital, and there is an optimal choice at any stage of the problem solution. It will produce the optimal solution for the complete problem to be solved. To make the greedy algorithm, it is essential to identify the optimal substructure in the problem to be addressed. Then it is also crucial to determine the entire requirement that the solution will include, thus the most significant sum and the shortest path possible.

## The Multilevel Paradigm for Processing Large Data Sets

The volume, variety, and velocity of the large data and the most valuable data it contains have boosted the morale of the investigation in the systems which process data. The approaches are always using database management systems (Ruiz & Stutzle, 2007). There is the use of map reduced pioneered, which is a paradigm change and is also serving as primary data and making it simple and more refined data. When the primary data is compared to DBMs, the map reduced will generate controversies in the processing system, which is more efficient, the level of abstraction, and the dataflow which is available.

## Conclusion

In conclusion, the big data system is nowadays standard and blooming, and this is inspired by map reduction. Most of the big data systems follow the plan-reduced system but with a very flexible model for general use. They also absorb the advantages of DBMSs with very higher abstraction. Some specific systems like machine learning and stream data processing and to explore such opportunities and in assisting users in the selection of suitable methods. The system benchmarks and the data flow will help in the use of the survey.

## References

Ahuja, R. K., Orlin, J. B., & Tiwari, A. (2000). A greedy genetic algorithm for the quadratic assignment problem. Computers & Operations Research, 27(10), 917-934.

Boese, K. D. (1995). Cost Versus Distance In the Traveling Salesman Problem (pp. 109-133). Los Angeles: UCLA Computer Science Department.

Gomez-Exposito, A., Abur, A., de la Villa Jaen, A., & Gomez-Quiles, C. (2011). A multilevel state estimation paradigm for smart grids. Proceedings of the IEEE, 99(6), 952-976.

Reinelt, G. (1994). The Traveling Salesman: Computational Solutions for TSP Applications. Springer-Verlag.Ruiz, R., & Stutzle, T. (2007). A simple and effective iterated greedy algorithm for the permutation flowshop scheduling problem. European Journal of Operational Research, 177(3), 2033-2049.

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