The Traveling Salesman Problem Computer Science Essay

The Traveling Salesman Problem Computer Science Essay Travelling sales man problem is one of the challenging problems in the real life and also most well studied combinatorial optimization problem. Many Researches from different fields like operational research, algorithms design and including artificial intelligence attract by it. This problem has been studied by different researches and come up with different solutions and this problem has been solved by using different algorithms like Blind search, Branch and Bound Search , Heuristic algorithm and Genetic algorithms etc. the problem was formulated as a mathematical problem in 1930 and later it is used as bench mark for many optimal solution. 1.1 TRAVELING SALESMAN PROBLEM: A Travelling salesman has a task of visiting N number of cities. He will start from a home location and want to visit each city just once and return back to the original location from where he starts. Travelling salesman route will be plan in such a way that in a given N number of cities cost of travelling from one city to any other city what is the minimum round trip route that visit each city once and then return to the starting place. The goal is to find the shortest tour that visit each city in a given cities exactly ones and then return to the starting city. The only solution to the travelling salesman problem is to calculate and compare the length of all possible ordered combinations. 1.2 History of travelling salesman problem: The travelling salesman problem was treated by a Irish mathematician sir William Rowan Hamilton and British mathematician Thomas Penyngton kirkman in the early 1800s. Hamilton and kirkman work on game called Hamilton Icosian game that requires player to complete tours through 20 points using only the specified connections. The general form of the travelling salesman problem studied by mathematician Karl menger during 1930s. He defines the problem using brute force algorithms and observed nearest neighbour heuristic non optimality. Soon after the name travelling salesman problem introduced by hassler Whitney at princeton university. In the 1940s the Travelling salesman problem was studied by statisticians Mahalanobiss, Jessen, Gosh, Marks. Among them P.C .Mahalanobiss took a sample survey of acreage under jute in Bengal discussed aspects of travelling salesman solutions through randomly chosen locations in the Euclidean plane. And this work is deal with survey of form lands one of the major cost to carrying out the survey was the transport ion of men equipment from one survey point to next. During the period between 1950s and 1960 the problem becomes more popular in scientific group in Europe and USA. a number of solutions designed by George B.Dantzing, Fulkerson and Johnson(1954) . in 1954 Heller publi shed an 88 report which contains many basic solutions on the travelling salesman polytype. In 1957 L.L.Barachet published graphic solution of travelling salesman problem which describes an enumeration scheme for computing near optimal tours. In 1964 R.L Karg and G.L. Thompson were applied heuristic algorithm for a 57 city problem. During the period R.M Karp and M.Held published an article about the travelling salesman and minimum spanning tree which introduced one tree relaxation of the travelling salesman problem and using node weights to improve the bound given by optimal tree. In 1990s Bixby and Cook developed the programme for travelling salesman problem which are using recently. 2.0 GRAPH: 2.1 Travelling salesman problem modelled as graph: Travelling salesman problem can be modelled as Graph where as the cities are the graph vertices, path is graph edges and path distance is edge distance. Our goal is to find the shortest tour that visits each city in a given graph exactly ones and then return to the starting city. 3.0 Search Algorithms: To solve travelling salesman problem we can use different types of algorithms like blind search, heuristic algorithms, uniform cost search etc 3.1 Blind search: Blind search algorithms are such algorithms which does not contain domain knowledge of the problem and it search blindly, hence it is called blind search algorithm. The only thing a blind search can do is it describes a non goal state from goal state. Assume that you are at city A and you want to reach at city D, if we draw a search tree level 1: city E, city B, city C. a blind search will have no idea which node should explore first, hence it explore each and every node blindly. In blind search we may not get information we can use. We just be looking for an answer and we wonà ¢Ã¢â€š ¬Ã¢â€ž ¢t no until we found it. The blind search is also called as un informed search. Blind search is divided into two searches Algorithm they are 1) Breadth first search and 2) Depth first search 3.2 Breadth first search: Breadth first search is a search which starts at one node and it expands the all the neighbour node then after completing those nodes, it expands remaining nodes unexpanded it process is continues up to reaching a goal. The technique followed by breadth first search is FIFO first in first out) queue, the difference between breadth first and depth first search is, the depth first uses stack i.e. LIFO (last in first out). In this logic the items which are added is equal to the item which are deleted. This process continues up to reach a goal. 3.3 Depth first search: Depth first search is a general technique for traversing a graph. The technique which organise the to Do list was stack that is last in first out (LIFO). Depth first search start at one node and it explore as far as possible along each branch until a required goal node is found. Then it takes backtracking and return to the next node which it hasnà ¢Ã¢â€š ¬Ã¢â€ž ¢t finished exploring and it keep on repeat the same procedure until it reaches to its goal. If Depth first search goes down a infinite branch and if it does not find a goal state or if it does not find the solution may be a better solution at a higher level in tree. Therefore depth first search is neither complete nor optimal. ALGORITHM: INPUT : A connected graph G ,a starting vertex 1 OUTPUT: An ordered spannig tree T of graph G with root vertex 1 Initialize tree as vertex 1 Initialize S as the set of proper edges incident on vertex 1 While s not equal to null Let e = dfs next edge(G,S). Let w be the non tree end point of edge e. Add edge e and vertex w to tree T. Update Frontier(G,S). Return Tree T. 4.0 DEPTH FIRST SEARCH APPLY TO GRAPH STARTING AT NODE 1: For the above tree diagram we are using Depth first search, from the graph the sales man starts at node a and visits each node and return to original node Step1: from node 1 there is a three ways to travel i.e. node 2, node3 and node5 in level 2 Step2: by following depth first search condition LIFO it expands node 2 until the goal reach So for node 2 there is a two ways to travel i.e. node5 and node4 there is only one way for sales man to reach node 3 so this is not minimal path to travel. After expanding first branch sales man reaches node 3 from node 3 sales man reaches node4 With path length of 8 and from node 4 he reaches node2 with path length of 5 and from node 2 sales man reaches node5 with path length of 6 then length of whole sub tree is 27. After expanding of second branch sales man expands node 5, from node 5 sales man travels node 2 with path length of 6 then from node2 sales man reaches node 4 with path length of 5 From node 4 sales man reaches node 3 path length of 8.then length of whole sub tree is 26 Here we follow the condition that sales man will not visit those not which have already visited. the left vertices in the graph are chosen before right, from above to roots the minimum spanning tree path is 1-5-2-4-3 =26.this is the best minimum root for sales man to visit each city started at city1. 4.1 DEPTH FIRST SEARCH APPLYING TO THE GRAPH STARTING AT NODE 2: For the above tree diagram we are applying depth first search on which sales man starting at node 2. From starting city we have three ways to travel i.e. node 1, node4 and node5. From node 1 sales man has again to two ways to travel i.e. node3 and node5.from node 3 it reaches node 4. but here if we want to visit node5 we have to backtrack again it will not satisfy the condition visiting a city once so we didnà ¢Ã¢â€š ¬Ã¢â€ž ¢t reach goal here sales man expands neighbour node 4, from node 4 sales man reaches node 3 with path length of 8, then sales man processed to node 1with path length of 8, then sales man visits node 5 from node 1 with path length of 7. The length of whole sub tree is 2-4-3-1-5=28. After expanding node4 travel sales man moves to node 5, from node 5 he reaches node1 path length of 7,from node 1 sales man reaches node 3 the path length of 8 and he proceeds to node 4 from node 3 with path length of 8,then whole length of sub tree is 2-5-1-3-4=29. Here sales man follow the condition that visiting each city, then the the best minimum cost route starting at node 2 is 2-4-3-1-5=28. 5.0 INFORMED SEARCH: Informed search is an estimate availability of the distance (cost) from each node(city) to the goal. This estimation will help you to head into the correct direction. The outline of inform search are heuristic search, best first search, greedy search and A* search 5.1 Heuristic search: Heuristic uses domain specific knowledge to estimate the quality or potential of partial solution .A heuristic search that find a good solution in less time comparing to blind search but not always find best solution. This search is very much useful for solving big problem which may may not solved by using other search and this search generate a possible solution which can be either route from the initial location or goal in the problem. For travelling salesman problem the nearest neighbour heuristic work well, but some time due to the arrangement of cites it will not find the shortest route. 5.2 Best first search: Best first search is a general approach of the tree search algorithm. In which it expand a nodes which is based upon a evaluation function. The estimation of cost is constructed as evaluation function then the node with minimum evaluation is expanded first. A best search is a combination of both breadth first search and depth search, a breath first is good because it will not go deeper and the depth first search is best because it can be found without searching all the nodes, hence the best first search allows us to gain benefit both the approaches 5.3 Greedy search: Greedy search select the path that has lowest heuristic value or estimated distance to the goal. Greedy search is a example of best first strategy, and it some cases like depth breast search I may never find the solution and greedy search is not optimal solution take into costly paths . This can be happen in the last step or in the first step. 5.4 A* search: The most widely known form of best first search is called a* search or best first search is simplified as A*search. The evaluation of nodes is carried in A* search was combination of g(n) the cost reached and h(n) the cost get from the node to the goal f(n)=g(n)+h(n) Here f(n) estimate cost of cheapest solution through node n, A* search is both complete and optimal which is identical to uniform cost search. From the above equation: g(n) is the total length between starting place to current location. h(n) is estimated length from current location to goal. A heuristic function which estimate how much distance it takes to reach goal. f(n) is the sum of g(n) and h(n) .then this will be a current estimated shortest path which founds until a A* algorithm is completed. A* Algorithm: A* Algorithm starts with initial nodes then it will take the best node on open such that f(n)=g(n)+h(n) is minimum. If Best is a goal node quit then return to the path from the intial to best or remove the best from open and all among best node naming each node with its path from initial node. 6.0 A* Algorithm applying for given graph: from the above graph , it shows that the node 1 is the initial node where salesman starts and reaches the node 3 which is the current location . then sum of the distance between initial to current location is consider as g (n) ie 1-5-2-4-3= 26.then the estimated length from current location to the goal is consider as h(n) ie 8 and then total minimal distance of path is A*=1-5-2-4-3-1=34 Then this is the minimal path in which salesman can travel starting at node 1. 7.0 Comparison table for depth first search and A* algorithm Algorithm Domain knowledge Memory space Minimum route Time complexity efficient Depth first search It doesnà ¢Ã¢â€š ¬Ã¢â€ž ¢t contain domain knowledge more It find after complete search It takes more time less A* search It contain domain knowledge less It has information about it It takes less time more 8.0 CONCLUSION: The Idea of Travel sales man problem has much application in different fields .To Find best routes of travelling salesman we are used two algorithms they are Depth First search and A* Algorithm. Hence we conclude that A* algorithm is more efficient then depth first search algorithm, the time complexity of depth first is more comparing to A* search and it takes less memory space comparing to depth first search. Depth first search algorithm is more suitable finding minimum tour for limited number of cities, because if we take 50 to 100 cites depth first search expands each node of a tree to reach to the goal which is time consuming and memory waste.

Role of Women in Hemmingways Hills like White Elephants, Lawrences Th

Role of Women in Hemmingway's Hills like White Elephants, Lawrence's The Horse Dealers Daughter and Faulkner's A Rose for Emily The role of women in society is constantly questioned and for centuries women have struggled to find their place in a world that is predominantly male oriented. Literature provides a window into the lives, thoughts and actions of women during certain periods of time in a fictitious form, yet often truthful in many ways. Ernest Hemmingway's "Hills like White Elephants", D.H. Lawrence's "The Horse Dealers Daughter" and William Faulkner's "A Rose for Emily" each paint a picture of a woman who has failed to break away from her male companion, all describing a stereotypically dominated woman. Through submissive natures, compliant attitudes, and shattered egos the three women each struggle to live their lives in accordance to men, using only silent means of escape. In Hemingway's "Hills Like White Elephants" we are introduced to Jig. Jig is a woman who lacks the ability to make decisions without constant approval and recognition from a man who has impregnated her but who would rather she aborts the baby. Jig, unfortunately, cannot make decisions on her own, which is exemplified throughout the story, depicting her weak and dependent personality. "What should we drink?" From the opening line of the story we are introduced to a character that questions rather than acts. Someone who is unsure of not only herself but the relationship she is involved in. Though a simple question about what beverage to order can often appear courteous, this is only the first of many examples pertaining to Jig's inability to live her life as an individual. Later she questions her purpose in life, "That's all we do isn't it-look... ... Emily's lack of ability to let any male dominant leave her life. She murdered this man so that she could have him with her at all times until she too could die. Emily never left her house, hiding from the world. Only in the company of a man would she venture out into the streets. Emily suffered from a lack of ability to let go and an uncontrollable desire to hold on to those men who would control her life. In all three stories women are represented in a tragic, pitiful light. In a society, those today supports women's abilities to be independent and live their lives as they chose once lived subordinate, dependent women. Stories such as these have helped to shape the present day's understanding of women's place in society. Works Cited  · Geddes, Gary, ed. The Art of Short Fiction. (pp.399-409) and (pp. 322-325)  · Faulkner, William. A Rose for Emily

Budgeting and Business Planning Essay

In order to create the criteria for the committee we have to look at what they want from the project. Most businesses and organisations are in business to make a profit, however the committee has different aims and objectives compared to a normal business or organisation. They need to weight up the options of each proposal and decide which best relates to their aims and objectives. When the committee is considering which proposal to go with they should consider the following categories †¢Short-term Financial Benefits †¢Long-term Financial Benefits †¢Environment †¢Relation to aims Short-term Financial Benefits The committee needs to consider the short term benefits of each proposal. They don’t want to take on a proposal that has high start-up costs and doesn’t make a return on the capital within the first year. The committee does not want to be left in a high amount of debt if the proposal fails. If the committee is left with a debt they may not be able to fund other projects to further their three aims. Long-term Financial Benefits Long-term the proposals will give different outcomes. The committee needs to decide on how long they want to keep moult hall? If they want it as a long term asset which can help with their aims? Or do they want a longer term money making scheme to boost revenue which can be used in areas they already control. Environment Moult Hall comes with a huge area of woodland, they needs to consider the effects on the local plant life and wildlife that any proposal could have. They need to think, does it destroy plant life? Does it impact on wildlife habitats? Is their going to be any long lasting environmental damage? Relation to Aims The committee is based on three main aims. They need to consider these aims in making their decision, they can’t be making use of one proposal if it goes strictly against some of their basic aims. They have to consider their own image and brand, does a proposal give them a bad name/image considering their aims and objectives? Question 2 Jonathan and Ingrid’s proposal has some short term financial benefits, in the 30 week trial run moult hall turns over a small profit of  £1,646. When the committee takes their proposal in to consideration they should recognise this point, moult hall will be of no financial burden in the short-term 30 week trial. If the 30 week trial is to succeed then moult hall can have some structural work done costing  £20,000 which would close moult hall for four weeks after the trial. The structural alterations would increase the capacity of the site to allow up to 30 guests to stay at one time. According to Jonathan and Ingrid projections of a weekly cost per guest of  £66 and a weekly charge per guest of  £150 they are making  £84 profit per guest per week. With the structural alterations and the increase in capacity by 20 guests. The weeks after the alterations are finished they could make an extra  £1,680 per week, bringing their possible weekly total profit at full capacity to  £2,436 (allowing for one free space, 29 paying customers). Therefore they could justify the large  £20,000 outlay on alterations as moult hall would pay for these alterations in little over 8 weeks. So in the long-run moult hall could be a profitable project by the end of year one. The one drawback of this proposal is the loss of the usage of the minibus at weekends, which generated an annual income of  £1,040, however it can be argued that it is now being put to better use and that the money is being recuperated from moult hall. Break-even point Total expenses £63,880 Weeks until break-even47.3 Guests until break-even473 As we can see from the table above moult hall will have to attract 473 guests a year to break even when they can only hold 10 guests per week. Equally they would have to run for 47.3 weeks a year at full capacity to break-even. This gives them an average of 9 guests a week. If they were to structurally improve moult hall so that they can hold up to 30 guests a week then the figures would look very different as shown in the table below. Break-Even Point Total Expenses £63,880 Weeks until break-even point14.7 Guests until break-even point441 Although a similar amount of guests is required the amount of weeks at full capacity has drastically dropped from 47.3 to 14.7. Their average guests per week has dropped from 9 to 8. Although this is not a large drop, in comparison they only have to fill 8 out of 30 beds compared to 9 out of 10 beds in the 30 week trial. The environment and its protection are very important to the committee and are mentioned in their main aims. Jonathan and Ingrid’s proposal helps the environment and makes the most of moult hall, if a garden was to be kept at moult hall the guest could tend to this and grow vegetables and recycle waste in a compost, which in turn could be re-used on the garden making moult hall very self-sufficient and environmentally friendly. One of the major aims of the committee is to help and educate the young. Jonathan and Ingrid’s proposal does just this, by inviting young people from the surrounding area to come stay and learn about the countryside. The guests will be able to learn about different wildlife and plant life living in the moult hall woodland and surrounding areas. The committee has to consider how the proposal will relate to their own aims, with moult hall becoming a learning centre for the young the committee could promote the good work they are doing to increase t heir donations revenue and grow as a trust. Question 3 Winston’s proposal consists of turning moult hall into a quad bike track with lavish bedrooms in the house for guests. In the year one moult hall will turn over a profit of  £1,034,283.  £750,000 of this is guaranteed to the North West trust for the protection of wildlife and the other  £284,283 will go to Winston himself. Moult hall would be a great money maker for the trust with the guaranteed income of  £750,000 adding to the  £800,000 a year they receive from local donations and fund raisers, boosting their total revenue for the year to around  £1,550,000. Long-term moult hall will turnover similar amount each year as long as there is no dip in demand. In the second year when Winston doesn’t have any initial capital costs he will make a large profit himself of  £402,350. The funds the committee will earn from moult hall could be used in other areas to promote the protection of wildlife. Break-even Point Total Expenses £1,022,650 Weeks until break-even point35.9 Guests until break-even point538 From the table above we can see the break-even point for moult hall under Winston’s proposal. He would need to be operating at full capacity for almost 40 weeks a year to break-even this means he would have to attract 538 guests a year. On average to break-even Winston will have to have 11 guests a week. In monetary terms moult hall will be a very successful; however some parts of the proposal will go against the trusts main aims. The trust was set up for the protection of wildlife; one aim is to protect local wildlife and plant life. To make the quad bike track many mature trees will have to be removed in the grounds of moult hall. This will disrupt some of the habitats of animals in the woodland. One of the main habitats that could be disrupted is the nesting sites of the red kite. The red kite has only recently been re-introduced in to the United Kingdom after the success of similar projects in wales; the trust fully supports the work of the national charity that achieved this. The red kite was wiped out in the UK by modern farming methods which use pesticides to kill small rodents, which are the main food source of the red kite. The first aim of the Trust is to encourage farming methods that don’t hurt local wildlife and plant life. They have to consider how the disruption of the nesting sites would reflect on them if they took on Winston’s proposal. It may look bad as with one hand they are supporting the work of the charity yet they are making money at the cost of disrupting local nesting sites. Question 4a There are many different measures that to committee can use to measure the performance of moult hall such as: †¢Monthly financial reports †¢Committee inspections twice a year †¢Variance analysis – comparing budgeted figures with actual Variance Analysis The committee could use variance analysis; variance analysis is a comparison of the budgeted cost of running moult hall and the actual cost of running moult hall. They could see if the costs of moult hall are favourable or adverse. If the results are favourable then this means that moult hall is running at a cheaper cost than they originally budgeted. However if the results are adverse then this means that the cost of running moult hall is more than they budgeted, this could be due to higher food prices or a larger light and heating bill than first anticipated. Variance analysis will give the committee a good idea of how much difference there is between their original planned budget and their actual outlay. This will be useful in determining weather moult hall is a financially viable option. Monthly Financial Reports The committee could ask for the manager of moult hall to send them monthly financial reports so that they can keep track of the performance and see if they are making or loosing money. They could let the financial reports come in for a few months or even up to year. This way they can identify trends and high and low seasons. They may find that they have a slightly seasonal product, as more people will want to be outside in the summer compared to the winter. They can also see if over a year they are getting an increasing amount of interest month on month or if they are losing interest Committee inspections Committee inspections could take place two or three times a year, the committee could travel as a whole or send a few representatives to moult hall to assess the upkeep of moult hall and the grounds. The inspection could also be used to see how the guests are enjoying their time at moult hall. The inspection team will then feed back to the committee who will have meetings on how to improve moult hall based on the feedback from guests and/or any improvements or checks that would need to be made to ensure the performance of moult hall is consistent. Question 4b When the committee is assessing the performance of moult hall they can use different companies to assess moult hall for them. The Environmental Inspection Agency (EIA) can carry out Environmental impact assessments. The committee could use this agency to assess the impact that moult hall is having on the surrounding woodland and grounds of moult hall. The committee can use this information to track the environmental progress of moult hall and see if the project is having a damaging or positive effect on the woodland and grounds. The committee could also use a survey company to produce a survey which can be given to guests when they leave so that the guests can give their feedback. The survey company can then use these results to produce accurate feedback to present to the committee. This method would be better than the committee asking the questions themselves as it will give a better representation of the guest’s views on moult hall. One company that they could use is amplitude research, this is a company that specialises in market research, Amplitude research can create a standard survey for guests which will make the results more comparable and easier for the committee to act upon.

The Threat Of The United States - 966 Words

We walk amongst each other weaving in and out of eachother’s lives. To our disadvantage, we all wear a distinct expiration date plastered upon our face. As a society, we struggle day to day fighting a battle in which we know the outcome of. To our dismay, time is not the enemy but our very own DNA is the true culprit. Today, we come together as a community to fight for our right at the longest life available. The aisles at our local pharmacies are packed full of rows and rows of anti-aging products that our consumer based culture is haunted by. Over the past century, the search to extend life has blown through the roof. We’ve become so preoccupied with the fear of death that society has tackled great lengths to combat the cycle of life. Much confusion has risen as to where the line falls in regards to what is appropriate and what is unethical. Recent findings have begun to emerge which are attracting light to the many flourishing possibilities that our science and technological advances offers. Numerous attempts are being made in the hopes of finding an answer to the problem of the millennium. The concept of cloning has caught scientists attention throughout this past century. The world has to come to grips that the cloning technology is almost here,† a representative for a European research group, Panos Zavos. Clones are life forms that have identical genetic coding. A clone is an organisms replica. Originating in the late 1880’s; the first record of synthetic embryoShow MoreRelatedThe Threat Of The United States844 Words   |  4 PagesUnited States While not geographically located in the South East Asian region, the US in recent years has implemented a policy referred to as the Asian Pivot. 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