Application of Transportation Theory to Scheduling a Military Tanker Fleet

A complete set of voyages for the tankers of the military fleet was computed so as to minimize the expected total distance to be traveled by the ships in ballast during the year ending 30 June 1950, based on deliveries according to the advance requirement estimates of the Armed Services Petroleum Purchasing Agency. The optimum route-schedule so computed required about 5 per cent less mileage in ballast than would a schedule consisting entirely of simple round trips. No comparisons have been made with actual operations to determine the extent of conformity between the computed schedule and the actual sailings. The object of the analysis was to test the analytical approach of Koopmans (Koopmans, Tjalling C., Stanley Reiter. 1951. A model of transportation. Tjalling C. Koopmans, ed. Activity Analysis of Production and Allocation . Proceedings of the Linear Programming Conference held at the University of Chicago, June 1949. Wiley, New York, 222--259.) on a fleet operation, as a first step in further development and application of such methods. The routeschedules obtained in this manner might well be compared with actual practice and the theory modified as necessary to allow for important factors not given proper attention in this first approximation. The author’s interest in the problem was aroused by papers on transportation theory presented by Koopmans (Koopmans, Tjalling C., Stanley Reiter. 1951. A model of transportation. Tjalling C. Koopmans, ed. Activity Analysis of Production and Allocation . Proceedings of the Linear Programming Conference held at the University of Chicago, June 1949. Wiley, New York, 222--259.) and Dantzig (Dantzig, George B. 1951. Application of the simplex method to a transportation problem. Tjalling C. Koopmans, ed. Activity Analysis of Production and Allocation . Proceedings of the Linear Programming Conference held at the University of Chicago, June 1949. Wiley, New York, 359--373.) at a conference on linear programing in Chicago during June 1949, under the auspices of The Cowles Commission for Research in Economics of The University of Chicago. Frank L. Hitchcock (Hitchcock, Frank L. 1941. The distribution of a product from several sources to numerous localities. J. Math. Phys 20 224--230.), L. Kantorovitch (Kantorovitch, L. 1942. On the translocation of masses. Doklady Akad. Nauk SSSR . 37 199--201.), and Tjalling C. Koopmans (Koopmans, Tjalling C. 1949. Optimum utilization of the transportation system. Econometrica 17 (Supplement) 136--146.) have contributed independent formulations of this same theory, and each has offered a computational scheme. Professor Koopmans, in his Introduction to the Chicago Conference Proceedings (Koopmans, Tjalling C., Stanley Reiter. 1951. A model of transportation. Tjalling C. Koopmans, ed. Activity Analysis of Production and Allocation . Proceedings of the Linear Programming Conference held at the University of Chicago, June 1949. Wiley, New York, 222--259.), has discussed the interrelation of the conference papers---including the bearing of some of these on the Hitchcock distribution problem. The present author (Flood, Merrill M. 1953. On the Hitchcock distribution problem. Pacific J. Math. 3 369--386.) has previously offered a computational scheme for treating the problem, including the troublesome degenerate cases. An extreme degenerate case is that known as the “personnel assignment problem,” as treated by Votaw and Orden (Votaw, D. F., A. Orden. 1952. The personnel assignment problem. Alex Orden, Leon Goldstein, eds. Symposium on Linear Inequalities and Programming . Project SCOOP, Hq. US Air Force. 155--163.) and von Neumann (John von Neumann. 1953. A certain zero-sum two-person game equivalent to the optimal assignment problem. H. W Kuhn, A. W. Tucker, eds. Contributions to the Theory of Games II . Princeton University Press, 5--12.) However, Orden has offered a simple and effective way to reduce the degenerate cases to nondegenerate ones easily manageable by Dantzig's (Dantzig, George B. 1951. Application of the simplex method to a transportation problem. Tjalling C. Koopmans, ed. Activity Analysis of Production and Allocation . Proceedings of the Linear Programming Conference held at the University of Chicago, June 1949. Wiley, New York, 359--373.) simplex method; consequently the shipping example treated in the present paper is typical computationally for the general Hitchcock distribution problem. Operations Research , ISSN 0030-364X, was published as Journal of the Operations Research Society of America from 1952 to 1955 under ISSN 0096-3984.