Selection of an Initial Solution for the Traveling-Salesman ProblemAuthor(s): Michael F. DaceySource: Operations Research, Vol. 8, No. 1 (Jan. - Feb., 1960), pp. 133-134Published by: INFORMSStable URL: http://www.jstor.org/stable/167549 .

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Letters to the Editor

SELECTION OF AN INITIAL SOLUTION FOR THE

TRAVELING-SALESMAN PROBLEM

Michael F. Dacey

University of Washington, Seattle, Washington

(Received October 19, 1959*)

T HE SOLUTION of the traveling-salesman problem recently given by CROES'1l has three phases: (a) the selection of an initial, feasible solution, (b) a pro-

cedure for improving the initial solution, and (c) the consideration of all further transformations to obtain the optimum solution. Algorithms by DANTZIG, FUL- KERSON, AND JOHNSON[2] and the intuitive approach due to BARACHETt3' contain similar phases. However, even for small order problems, these procedures are generally laborious. The availability of a simple, noniterative procedure for the selection of initial, near-optimum solutions is, apparently, desirable. These initial solutions may suffice in the practical situation, or they may be used as the starting point in an iterative procedure, such as the one described by Croes.

The present writer has developed a procedure that is not iterative and that rapidly selects a single solution. Experience in solving a number of problems by this method discloses a persistent, though not necessary, tendency for these solu- tions to approach optimum. Full details of this procedure are in a mimeographed "Discussion Paper" by DACEYJ41

In this note, only the frequency of the selection of optimum and near-optimum solutions by this procedure is discussed. A comparison is made between these initial solutions of traveling-salesman problems and their optimum solutions. Comparative examples have been chosen from well-known papers on this subject.

The procedure was used to obtain initial solutions for the ten problems of order nine solved by ROBACKER.[53 The optimum solution is selected for five of these problems, his Tables 1, 3, 6, 8, and 10. The average distance of the ten initial solutions is 4.8 per cent longer than optimum. Five minutes, on an average, were taken to obtain each solution. Starting from a random, initial solution and using

* The complete paper (reference 4) was submitted for publication in OPERA- TIONS RESEARCH on March 17, 1959. On the advice of the referees, the author was requested to prepare this Letter summarizing his work and calling attention to the availability of the complete paper in mimeographed form. This request was based on the feeling that only a limited number of readers would be interested in the details of Dacey's specialized techniques for the selection of an initial solution; but that, on the other hand, the availability of these techniques should be a matter of record-EDITOR.

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134 Letters to the Editor

the method proposed by Dantzig, Fulkerson, and Johnson, Robacker required an average of three hours to solve these problems.

The initial solution obtained by this writer for the problem given by Croes has a cost of 265; this is 7.2 per cent greater than the optimum solution of 246. For the problem given by Barachet, and recently used by DANTZIG, FULKERSON, AND JOHNSON,[f1i the initial solution is 3.2 per cent longer than the optimum dis- tance. In both cases, the use of the approximative procedure saved a substantial amount of time.

To some degree these results corroborate the efficacy of the procedure for select- ing near-optimum solutions for small-order matrices. Similar results are obtained for larger-order matrices. When the distances between cities have an empirical reference, the frequency of selection of optimum solutions is greater, and the aver- age error tends to be smaller. To determine an initial solution for the 42-city case given by DANTZIG, FULKERSON, and JOHNSON[2) required about 90 minutes and it was 4 per cent longer than the optimum distance.

REFERENCES

1. G. A. CROES, Opns. Res. 6, 791-812 (1958). 2. G. B. DANTZIG, D. R. FULKERSON, AND S. M. JOHNSON, Opns. Res. 2, 393-410

(1954). 3. L. L. BARACHET, Opns. Res. 5, 841-845 (1957). 4. M. F. DACEY, "Selection of an Initial Solution for the Traveling-Saleslan

Problem," Mimeographed for private distribution, available from the De- partment of Geography, University of Washington (1959).

5. J. T. ROBACKER, "Some Experiments of the Traveling-Salesman Problem," Rand Report RM-1521, 1955.

6. G. B. DANTZIG, D. R. FULKERSON, AND S. M. JOHNSON, Opns. Res. 7, 58-66 (1959).

DESIGN OF THREAT MODELS*

Harold Tombacht

System Development Corporation, Santa Monica, California

(Received June 3, 1959)

THE PURPOSE of this Letter is to formalize an approach to threat model design that provides the model builder with criteria for selecting, from the

universal sot of all possible threat models, a limited subset of models that are realistic, usable, and useful. It is believed that this approach will contribute to the elimination to a large degree of the arbitrariness that often accompanies the

* This is a condensed version of a paper presented at the Fifteenth National Meeting of the OPERATIONS RESEARCH SOCIETY OF AMERICA, Washington, D. C., May 14-15, 1959.

t Now with Hughes Aircraft Company, Culver City, California.

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