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Sequencing a One State-Variable Machine: A Solvable Case of the Traveling Salesman Problem

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  • P. C. Gilmore

    (IBM Watson Research Center, Yorktown Heights, New York)

  • R. E. Gomory

    (IBM Watson Research Center, Yorktown Heights, New York)

Abstract

We consider a machine with a single real variable x that describes its state. Jobs J 1 , …, J N are to be sequenced on the machine. Each job requires a starting state A , and leaves a final state B i . This means that J i can be started only when x = A i and, at the completion of the job, x = B i . There is a cost, which may represent time or money, etc., for changing the machine state x so that the next job may start. The problem is to find the minimal cost sequence for the N jobs. This problem is a special case of the traveling salesman problem. We give a solution requiring only 0( N 2 ) simple steps. A solution is also provided for the bottleneck form of this traveling salesman problem under special cost assumptions. This solution permits a characterization of those directed graphs of a special class which possess Hamiltonian circuits.

Suggested Citation

  • P. C. Gilmore & R. E. Gomory, 1964. "Sequencing a One State-Variable Machine: A Solvable Case of the Traveling Salesman Problem," Operations Research, INFORMS, vol. 12(5), pages 655-679, October.
    • Handle: RePEc:inm:oropre:v:12:y:1964:i:5:p:655-679
    DOI: 10.1287/opre.12.5.655
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