On Some Works of Kantorovich, Koopmans and Others

The July, I960, issue of Management Science contains an English translation of an important original article by L. V. Kantorovich [Kantorovich, L. V. Mathematical Methods of Organizing and Planning Production. Leningrad University, 1939, with a Foreword by A. R. Marchenko (Russian). An English translation, prepared by R. W. Campbell and W. H. Marlow, appears under this same title in Management Science Vol. 6, No. 4, (July, 1960), pp. 366-422.] and an introductory note by T. C. Koopmans [Koopmans, T. C. 1960. A note about Kantorovich's paper, `Mathematical Methods of Organizing and Planning Production'. Management Sci. 6 (4, July) 363-365.] which is both evaluative and explanatory. In his penultimate paragraph, Professor Koopmans accords a tribute to this writing of Kantorovich which is at least partly deserved, although it is also paid for, we think, at too high a price in modesty for Professor Koopman's own work as well as the work of others. There is a certain lack of precision at critical points in the Kantorovich article which may tend to compound this price if allowed to stand through time. Professor Koopmans does not take adequate account of this lack of precision--e.g., in his discussion of the computing methods--perhaps because he restricted himself to a very short preface. Also, he runs together various major ideas (resolving multipliers, duality and efficiency prices) which need to be separated for more careful examination. Finally, the discussion is so cryptic as to raise some questions as to what Koopmans had in mind in his closing references wherein he compares Kantorovich's accomplishments (in the instant article) to those of von Neumann, Dantzig, et al. In this paper we shall examine some of these ideas at greater length than Koopmans permitted himself but without necessarily confining ourselves to the instant article by Kantorovich or to its preface by Koopmans.