handout2spring

handout2spring - ENGRI 1101 Engineering Applications of OR...

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ENGRI 1101 Engineering Applications of OR Spring ’10 Handout 2 The traveling salesman problem The traveling salesman problem is one of the most notorious optimization problems. The setting for the problem is as follows. A salesman starts at his home and has a given set of cities to visit. That is, if his home is in NY, and he must visit Syracuse, Chicago, San Francisco, Los Angeles, Detroit, and Pittsburgh, then one possible solution is to go from NY to Chicago to San Francisco to Los Angeles to Detroit to Pittsburgh to Syracuse, and then to return home to NY. Given that he is traveling to many cities and not staying over a Saturday night, he only quali±es for the standard coach airfare between each consecutive pair of cities that he visits. He knows the airfare between each pair of cities. That is, in our example, he might be given the following table of airfares: NY Syracuse Chicago SF LA Detroit Pittsburgh NY 202 135 245 245 169 129 Syracuse 202 309 445 445 230 160 Chicago 135 309 180 180 105 120 SF 245 445 180 39 195 165 LA 245 445 180 39 195 165 Detroit 169 230 105 195 195 135 Pittsburgh 129 160 120 165 165 135 Notice that the data contain a few irregularities (common to the airline industry). The airfare from Syracuse to SF is $445, but the airfare from Syracuse to Pittsburgh is $129 and the airfare from Pittsburgh to San Francisco is $165, for a total fare of $294. Unlike in this case, if the table of fares has the property that the cheapest way to go between each pair of cities is to take the non-stop flight, then the fares are said to satisfy the triangle inequality . In cases such as the one above, they are said to violate the triangle inequality. We shall assume that the salesman is pressed for time, and always takes the non-stop flight between each of his stops on his tour. Thus, the cost of the tour proposed above is: 135 + 180 + 39 + 195 + 135 + 160 + 202 = 1046 . Furthermore, if he were to go from NY to Syracuse to San Francisco to LA to Chicago to Detroit to Pittsburgh to NY, then the cost would be 202 + 445 + 39 + 180 + 105 + 135 + 129 = 1235 . Clearly, the ±rst tour is better. The salesman would like to choose the order in which to visit the cities so as to minimize the total cost of his trip. For the data given above, is the ±rst proposed tour the cheapest one? The traveling salesman problem is an optimization problem . For any optimization problem, there is some notion of what kind of input is expected. For the traveling salesman problem, the input consists of a table of costs, such as the one given above, and it could involve any number of cities. For any optimization problem, there is also a notion of a feasible solution ; that is, a possible answer (though not necessarily the best answer). For this problem, a feasible solution is a tour that visits all of the cities and returns to the starting point. And ±nally, there is the notion of an objection function , the criterion by which we choose which of the feasible solutions is the best one, the optimal solution . In this case, the objective function is the sum of the costs of flying between each pair of cities that occurs consecutively in the tour. In this case, our objective function is to
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This note was uploaded on 09/07/2011 for the course OR 1101 taught by Professor Trotter during the Fall '09 term at Cornell.

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handout2spring - ENGRI 1101 Engineering Applications of OR...

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