M.M Flood. The traveling salesman problem. Operational Research, pages 61--75, 1956.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....4 1or4# 3. This information is contained in cell R 43 of the reward matrix. Similarly, node 1 can transmit in step 2 if the transmission chosen in step 1 is 1 and node 2 can transmit in step 2 if the transmission A similar auxiliary variable formulation for the TSP was suggested by Flood [6]. Sequentiality here means that if node i is the transmitting node in the kth step of the solution, it must have been reached by any of the transmissions upto step k chosen in step 1 is either 4 1or4# 2or4# 3. We can thus set up the node transmission blocking constraints for step 2 as ....

.... that no upper bound is required to be declared for the integer variables X ijk as it is set implicitly by equations (23) 24) and (25) The number of variables and constraints in this formulation are both of the order O(N ) assuming = N 1) similar to Flood s IP formulation of the TSP [6]. B. Obtaining sub optimal solutions by limiting k We mentioned in Section VI that the node transmission blocking constraints and step transmission forcing constraints need to be repeated for all steps 2 in a broadcast application. This is necessary to obtain the optimal solution. A ....

) Flood Merrill M., 1956-1, "The Traveling Salesman Problem," in J. F. McCloskey and J. M. Coppinger (eds), Operations Research for Management, Vol. II, Johns Hopkins Press, Baltimore, Maryland, 1956, pp. 340-357.

....[11] as benchmark instances: BERLIN52, 52 locations in Berlin (Groetschel) CH130, a 130 city problem (Churritz) and A280, drilling problem (Ludwig) The optimum solutions for these instances are known. The generation of neighbours is done according to the well known Lin 2 Opt operator [4,6]. The second problem is the optimisation of a continuous function of real variables. We have chosen the function F6 2 (see [12] This problem instance is an interesting benchmark because of the great number of local optima and the distance between them. Moreover, the hole around the optimum is ....

M.M. Flood, The traveling salesman problem, Operations Research 4 (1956), 61--75.

....A local search heuristic that uses the k Opt neighborhood is usually called simply k Opt, and in this section we study various pure and restricted heuristics of this kind. Currently, 2 Opt and 3 Opt are the main k Opt heuristics used in practice, introduced respectively by Flood and Croes [27, 24] and by Bock [15] In Shen Lin s influential 1965 study of 3 Opt [49] he concluded that the extra time required for 4 Opt was not worth the small improvement in tour quality it yielded, and no results have appeared since then to contradict this conclusion. In contrast, there have been several ....

M.M. Flood. The traveling-salesman problem. Oper. Res., 4:61--75, 1956.

....according to the following rule: 0 1 = ij ij (3) where 0 is the initial value of trails. It was found that ) 1 0 h J n = is a good value for this parameter where h J is the length of the initial solution produced by the nearest neighbor heuristic (Flood, 1956) and n is the number of nodes. An interesting aspect of the local updating is that while edges are visited by ants, Equation 3 makes their trail diminish, making them less and less attractive, and favoring therefore the exploration of not yet visited edges and diversity in solution generation. ....

M. M. Flood, The Traveling Salesman Problem, Operations Research 4, 1956, 61-75.

....of the convex hull of the n cities, and the other m cities lie on a line segment inside this convex hull. This special case of the n city Euclidean TSP will be called the convex hull and line TSP. Awell known result with respect to the Euclidean TSP, presumed to be #rst mentioned explicitly by Flood #1956#, states that in the euclidean plane the minimal #or optimal# tour does not intersect itself . An intersection of a tour # is de#ned as a common point v 62 fp 1 ; p n g that is shared bytwo #or more# edges of #,or a common point w 2fp 1 ; p ng that is shared by three #or more# edges of ....

Flood, M. M. #1956#, The traveling-salesman problem, Operations Research 4, 61# 75.

....of the convex hull of the n cities, and the other m cities lie on a line segment inside this convex hull. This special case of the n city Euclidean TSP will be called the convex hull and line TSP. A well known result with respect to the Euclidean TSP, presumed to be first mentioned explicitly by Flood [1956], states that in the euclidean plane the minimal (or optimal) tour does not intersect itself . An intersection of a tour is defined as a common point v 62 fp 1 ; p n g that is shared by two (or more) edges of , or a common point w 2 fp 1 ; p n g that is shared by three (or more) ....

Flood, M. M. (1956), The traveling-salesman problem, Operations Research 4, 61-- 75.

....according to the following rule: 0 1 t r t r t = ij ij (3) where t 0 is the initial value of trails. It was found that ) 1 0 h J n y t = is a good value for this parameter where h J y is the length of the initial solution produced by the nearest neighbor heuristic (Flood, 1956) and n is the number of nodes. An interesting aspect of the local updating is that while edges are visited by ants, Equation 3 makes their trail diminish, making them less and less attractive, and favoring therefore the exploration of not yet visited edges and diversity in solution generation. 5.4 ....

M. M. Flood, The Traveling Salesman Problem, Operations Research 4, 1956, 61-75.

....or even superior to EAs. In the field of combinatorial optimization, local search algorithms (LSAs) have a long history since they are intuitive and very efficient. For example, the first local search algorithm for the traveling salesman problem has been proposed in 1956 58 (Croes, 1958; Flood, 1956), and a local search for the facilities location problem has been developed before 1962 (Armour and Buffa, 1963) Fig. 1 shows the general local search algorithm for a maximization problem: Beginning with a feasible solution to the problem, a new solution with a higher fitness f is searched in its ....

Flood, M. M. (1956). The Traveling--Salesman Problem. Operations Research, 4:61--75.

....with some simple special cases that result from convex sets. Section 3.2 deals with the case where the cities lie on a small number of line segments and Section 3.3 deals with the necklace TSP. 3. 1 Special cases related to the convex TSP In 1956, in one of the first papers on the TSP, Flood [49] observed that in the Euclidean plane the minimal TSP tour does not intersect itself . This observation is an immediate consequence of the quadrangle inequality (see Monge [76] 1781) which states that in a convex planar quadrangle p 1 p 2 p 3 p 4 d(p 1 ; p 2 ) d(p 3 ; p 4 ) d(p 1 ; p 3 ) ....

....generality assume that all k lines are horizontal. We order the points along every line from left to right, i.e. a OE b if a lies on the same line and to the left of b. Consider the shortest tour . Obviously, cannot contain long chords since then it would intersect itself and contradict Flood [49]. It can also be shown that if a subset of the edges in would form a fence, then had to intersect itself. Summarizing, the shortest tour is long chord free and fence free and Theorem 3.2 applies. In the convex hull and line TSP, the first class of cities is ordered along the upper chain of ....

M.M. Flood, The traveling salesman problem, Operations Research 4, 1956, 61--75. REFERENCES 51

....parts of g(s 1 ) and g(s 2 ) and some other curves) This is a contradiction and completes the proof. In the next section, we will also apply the following result that apparently has first been formulated by Flood in 1957 and that since then has become TSP folklore. Proposition 4. 4 (Flood [4]) In the Euclidean plane, the shortest TSP tour does not intersect itself (an intersection is either a city shared by three or more edges, or a non city point shared by two or more edges) 5 The Convex Hull and k Line TSP This section gives an exact definition of the convex hull and k line TSP, ....

M.M. Flood, The traveling salesman problem, Operations Research 4, 1956, 61--75.

....Supnick [12] Another important special case of the TSP is the Euclidean TSP: Here the cities are points in the two dimensional plane and their distances are measured according to the Euclidean metric. It is easy to see that in this case, the shortest TSP tour does not intersect itself (cf. Flood [4]) and hence, geometry makes the problem somewhat easier. Nevertheless, this special case is still NP hard (see e.g. Papadimitriou [6] or chapter 3 in the TSP book [8] The subject of this paper is to identify easy instances of the Euclidean TSP based on the concept of Demidenko (Kalmanson, ....

M.M. Flood, The traveling salesman problem, Operations Research 4, 1956, 61--75.

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M.M Flood. The traveling salesman problem. Operational Research, pages 61--75, 1956.

No context found.

M.M Flood. The traveling salesman problem. Operational Research, pages 61--75, 1956.

No context found.

M. M. Flood, "The Traveling--Salesman Problem," Operations Research, vol. 4, pp. 61-- 75, 1956.

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