Inventory Theory

W to begin? I became involved with inventory theory in 1955, but the proper starting point for this memoir is some years earlier, in the fall of 1951, when I began my graduate training in the Department of Mathematics at Princeton University. I had lived at home during my undergraduate years at Temple University and was totally unprepared for the remarkable features of life at Princeton in the early 1950s. I had a room in the splendidly gothic Graduate College, and in a short time I met my classmates Ralph Gomory, Lloyd Shapley, John McCarthy, Marvin Minsky, Serge Lang, and John Milnor. John Nash and Harold Kuhn had left Princeton the year before, but I saw a good bit of them during their regular returns. Martin Shubik was then a graduate student in the Department of Economics, passionately engaged with Oscar Morgenstern in the early development of game theory. Some 50 years later, Martin and I have offices in the same building, the Cowles Foundation, at Yale University, and Ralph and I meet regularly. I wrote my Ph.D. thesis under the direction of Salomon Bochner, a student of Erhard Schmidt, who was himself a student of David Hilbert. Albert Tucker was the chairman of the department; other faculty members were Soloman Lefschetz, William Feller, Emil Artin, and Ralph Fox. I got to know John Tukey during a daily commute to Bell Labs in the summer of 1953, where I would occasionally have a glimpse of Claude Shannon. I heard von Neumann lecture on computers and the brain and would often see Einstein and Gödel during their regular afternoon walks near the Institute for Advanced Study. I left Princeton for the RAND Corporation in June of 1954. One of my reasons for choosing RAND rather than a more conventional academic appointment was my desire to be involved in applied rather than abstract mathematics. I could not have selected a better location to achieve this particular goal. George Dantzig had arrived recently and was in the process of applying linear programming techniques to a growing body of basic problems. Richard Bellman was convinced that all optimization problems with a dynamic structure (and many others) could be formulated fruitfully, and solved, as dynamic programs. Ray Fulkerson and Lester Ford were working on network flow problems, a topic that became the springboard for the fertile field of combinatorial optimization. Dantzig and Fulkerson studied the traveling salesman problem and other early examples of what ultimately became known, under the guidance of Ralph Gomory, as integer programming. In 1955, the organization was visited by a budgetary crisis and I was asked if I would mind taking up temporary residence in the Department of Logistics. The Logistics Department was a junior subgroup of the Department of Economics at the RAND, with a much more prosaic mission than that of its senior colleagues. The members of the Logistics Department were concerned with scheduling, maintenance, repair, and inventory management, and not with the deeper economic and strategic questions of the Cold War. I moved into a simple office, far from my previous colleagues in mathematics, and sat for a few weeks wondering what I was meant to do. I don’t remember receiving any specific instruction or being presented with any particular research topic, but at some point I learned about the most elementary inventory problem: The decision about the quantity of a single nondurable item to purchase in the face of an uncertain demand. In the terminology of my first paper on inventory theory, “A Min-Max Solution of an Inventory Problem,” the marginal cost of purchasing the item is a constant c. If y units are purchased and the demand is , then the actual sales will be min y and if the unit sales price is r , profits will be given by the random amount,