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Inventory Theory INVENTORY THEORY BY HERBERT E. SCARF COWLES FOUNDATION PAPER NO. 1036 COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY Box 208281 New Haven, Connecticut 06520-8281 2002 http://cowles.econ.yale.edu/ INVENTORY THEORY HERBERT E. SCARF Sterling Professor of Economics, Yale University, 88 Blake Road, Hamden, Connecticut 06517, [email protected] here to begin? I became involved with inventory the traveling salesman problem and other early examples Wtheory in 1955, but the proper starting point for of what ultimately became known, under the guidance of this memoir is some years earlier, in the fall of 1951, Ralph Gomory, as integer programming. when I began my graduate training in the Department of In 1955, the organization was visited by a budgetary Mathematics at Princeton University. I had lived at home crisis and I was asked if I would mind taking up tempo- during my undergraduate years at Temple University and rary residence in the Department of Logistics. The Logis- was totally unprepared for the remarkable features of life tics Department was a junior subgroup of the Department at Princeton in the early 1950s. I had a room in the of Economics at the RAND, with a much more prosaic splendidly gothic Graduate College, and in a short time I mission than that of its senior colleagues. The members met my classmates Ralph Gomory, Lloyd Shapley, John of the Logistics Department were concerned with schedul- McCarthy, Marvin Minsky, Serge Lang, and John Mil- ing, maintenance, repair, and inventory management, and nor. John Nash and Harold Kuhn had left Princeton the not with the deeper economic and strategic questions of the year before, but I saw a good bit of them during their Cold War. regular returns. Martin Shubik was then a graduate stu- Imoved into a simple office, far from my previous col- dent in the Department of Economics, passionately engaged leagues in mathematics, and sat for a few weeks wonder- with Oscar Morgenstern in the early development of game ing what I was meant to do. I don’t remember receiving theory. Some 50 years later, Martin and I have offices in any specific instruction or being presented with any par- the same building, the Cowles Foundation, at Yale Univer- ticular research topic, but at some point I learned about sity, and Ralph and I meet regularly. the most elementary inventory problem: The decision about I wrote my Ph.D. thesis under the direction of the quantity of a single nondurable item to purchase in the Salomon Bochner, a student of Erhard Schmidt, who was face of an uncertain demand. In the terminology of my himself a student of David Hilbert. Albert Tucker was first paper on inventory theory, “A Min-Max Solution of the chairman of the department; other faculty members an Inventory Problem,” the marginal cost of purchasing were Soloman Lefschetz, William Feller, Emil Artin, and the item is a constant c.Ify units are purchased and the Ralph Fox. I got to know John Tukey during a daily com- demand is , then the actual sales will be miny and if mute to Bell Labs in the summer of 1953, where I would the unit sales price is r, profits will be given by the random occasionally have a glimpse of Claude Shannon. I heard amount, von Neumann lecture on computers and the brain and r miny− cy would often see Einstein and Gödel during their regular afternoon walks near the Institute for Advanced Study. The standard treatment of the problem was to assume a I left Princeton for the RAND Corporation in June of known probability distribution for demand, with the cumu- 1954. One of my reasons for choosing RAND rather than a lative distribution given by ,sothat expected profits more conventional academic appointment was my desire to are be involved in applied rather than abstract mathematics. I could not have selected a better location to achieve this par- r minyd− cx ticular goal. George Dantzig had arrived recently and was 0 in the process of applying linear programming techniques It is trivial to set to 0 the derivative of expected profit as to a growing body of basic problems. Richard Bellman was a function of y and obtain the optimal quantity to be pur- convinced that all optimization problems with a dynamic chased as the solution of the equation structure (and many others) could be formulated fruit- 1 − y = c/r fully, and solved, as dynamic programs. Ray Fulkerson and Lester Ford were working on network flow problems, a In my paper, the probability distribution of demand is topic that became the springboard for the fertile field of assumed not to be fully known. I study the decision prob- combinatorial optimization. Dantzig and Fulkerson studied lem in which the inventory manager selects the inventory Subject classifications: Inventory/production: personal reflections. Professional: comments on. Area of review: Anniversary Issue (Special). Operations Research © 2002 INFORMS 0030-364X/02/5001-0186 $05.00 Vol. 50, No. 1, January–February 2002, pp. 186–191 186 1526-5463 electronic ISSN Scarf / 187 level y that maximizes the minimum expected profit for former case the sale is lost, and in the latter case the sale all probability distributions of demand with a fixed mean is backlogged. and standard deviation . There is no particular reason The paper has two major results. The first is to show for thinking in advance that this version of the inventory that when sales are backlogged, the optimal ordering pol- problem would have anything but a clumsy solution, but in icy is a function of the stock on hand plus the stock pre- fact the answer turned out to be surprisingly simple. If we viously ordered and not yet delivered. The second result is define the function to show that policies of this simple form are not optimal 1 − 2a when sales are lost. A more detailed study of optimal poli- fa= 1/2 cies is then presented for the case of lost sales, a delay in a1 − a the receipt of orders of a single period, and a purchase cost then the optimal policy is to stock strictly proportional to the quantity ordered. I remember an early conversation on this topic with Harry Markowitz, + f c/r if 1 + 2/2<r/c which took place in a rowboat, in the ocean off the coast y = 0if1 + 2/2>r/c of Santa Barbara, as we watched a seal sunning itself on a marker buoy. The paper continues with a comparison between this inven- The analysis of optimal policies in the case of lost tory policy and those associated with the normal and Pois- sales is conducted by means of the standard dynamic pro- son distributions. gramming formulation of the inventory problem. Let the It was my good fortune to meet Samuel Karlin and inventory on hand at the beginning of the period be x, and Kenneth Arrow at RAND. They were both interested in suppose that the initial decision is to order up to y units at inventory problems, and they kindly invited me to spend a cost of cy − x.Ifthe time lag is a single period, the the academic year 1956–1957 with them at Stanford Uni- order will be delivered at the end of the period, resulting versity. My natural home at Stanford would have been a in a new level of inventory maxy − 0, where is the department of operations research, but such a department random demand for stock during the period, governed, say, had not yet been established, and I was formally located in by a probability distribution with density . Let Ly x the Department of Statistics. My office was in a charming be the expected costs experienced during the period, and building known as Serra House, sitting in a grove of euca- the discount factor. Then fx, the discounted expected lyptus trees at the edge of the Stanford campus. I was on costs associated with the series of optimal decisions, will the second floor of the building along with Kenneth, Hiro satisfy the dynamic programing equation Uzawa, and Patrick Suppes. Richard Atkinson was on the first floor, and Leo Hurwicz and Bill Estes were frequent visitors. fx= Min cy − x+ Ly x yx Kenneth and Sam became good friends and mentors; y both had an enormous impact on my professional career. + f0 d + fy− d We worked intensively on inventory problems during this y 0 year, and our efforts resulted in a monograph entitled “Studies in the Mathematical Theory of Inventory and Pro- If delivery were instantaneous, the expected costs during duction,” published in 1958. I will describe two papers that the period would be a function, Ly,ofthe immediately appeared in the monograph, in addition to the previously available inventory; and if backlogging were permitted, the cited min-max solution. stock level could become negative and the pair of integrals The first of these, entitled “Inventory Models of the in braces would be replaced by the single integral Arrow-Harris-Marschak Type with Time Lag,” was co- authored with Samuel Karlin. It was concerned with an fy− d important feature of inventory problems: that an order, 0 when placed, may not be immediately delivered. Two dis- resulting in the somewhat simpler equation tinct treatments of the problem caused by time lags in delivery are examined. Imagine, for concreteness, a retailer fx= Min cy − x+ Ly + fy− d whose stock is replenished by purchases from a wholesaler. yx 0 When a customer arrives at the retailer with a request for the item, the currently available stock may be insufficient The classic work of the three authors Arrow, Harris, and to meet this demand, even though there may be an adequate Marschak, whose names appear in the title of the paper, quantity previously ordered and sitting in the pipeline.
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