This paper describes a research strategy and its results, which guided a 40+ year collaboration between Abraham Charnes and William W. Cooper, which was initiated in a research center established at the (then) new Graduate School of Industrial Administration at Carnegie Institute of Technology (now Carnegie Mellon University). Initiated in collaboration with Bob Mellon of Gulf Oil Company, this strategy involved on-site collaborations with company personnel in more than 100 different companies and government agencies. An appendix to this paper describes the efforts of another team working from the same research center. This team, consisting of Charles Holt, Franco Modigliani, John Muth, and Herbert A. Simon, used a different research strategy with an accompanying application that also had important impacts on operations research/management science and other disciplines.

Chance constrained programming admits random data variations and permits constraint violations up to specified probability limits. Different kinds of decision rules and optimizing objectives may be used so that, under certain conditions, a programming problem (not necessarily linear) can be achieved that is deterministic—in that all random elements have been eliminated. Existence of such “deterministic equivalents” in the form of specified convex programming problems is here established for a general class of linear decision rules under the following 3 classes of objectives (1) maximum expected value (“E model”), (2) minimum variance (“V model”), and (3) maximum probability (“P model”). Various explanations and interpretations of these results are supplied along with other aspects of chance constrained programming. For example, the “P model” is interpreted so that H. A. Simon's suggestions for “satisficing” can be studied relative to more traditional optimizing objectives associated with “E” and “V model” variants.

A relatively robust method for the approximate analysis of the mean queue length of an M/G/c queueing system is proposed. The approximation method is developed based on the following assumptions: the residual service time of one busy server is independent of those of the other busy servers, and the system in which all the servers are busy is treated in the same way as a single-server system with c times the service rate of one of the servers. The application of these two assumptions is coupled through the introduction of a parameter n p . If the number of customers in the system is larger than n p , assumption 2 is used; otherwise assumption 1 is used. We found that certain properties of n p allow an estimation of the mean queue length of a large M/G/c queueing system through the approximate analysis of the mean queue length of a much smaller M/G/c queueing system. Numerical results show that the approximation is accurate even when the coefficient of variation of the service time and the number of channels of the system are as large as 20 and 200, respectively.

This paper examines properties and interrelations of several concepts of generalized concavity. It shows that the class of functions that are both “generalized concave” and “generalized convex” is closely related to the class of monotone functions of a single variable. After excluding a certain small class of exceptions, the paper shows that, for arbitrary (perhaps not differentiable) functions, concave implies pseudoconcave, pseudoconcave implies strictly quasiconcave, and strictly quasiconcave implies quasiconcave. Several results characterizing the extreme values of generalized concave functions are given.

This paper describes a research strategy and its results, which guided a 40+ year collaboration between Abraham Charnes and William W. Cooper, which was initiated in a research center established a...

In a paper published in Opns. Res. 7, 483–498 (1959), F. Hanssmann puts forward a model for inventory control of single stores, unbranching chains of stores, and finally branching chains of stores. In particular the model can be used for the problem of a central store supplying sub stores. Hanssmann's model is unrivalled in its versatility, but is derived in an intuitive, heuristic manner, and is open to some criticism on these grounds. The present author has tried to correct these deficiencies and in most cases has verified Hanssmann's intuition. A few of Hanssmann's results appear to be inaccurate, however, and alternatives are put forward to replace these.

Chance-constrained programming methods are applied to examine some statistical properties of PERT networks. Using duality, the PERT method is shown to be equivalent to use of the crudest linear decision rule and the confidence (or lack thereof) in meeting constraints is explicitly presented. The distribution of completion times (= Tintner's stochastic programming) follows easily and may often be multimodal, contrasting with (erroneous) central limit theorem usages in the literature. Possible extensions and developments of PERT using more adequate chance-constrained models and techniques are suggested and will be presented elsewhere.

Model equivalences may sometimes be used to replace “realistic” but unwieldy initial formulations with simpler counterparts. This can involve sophisticated uses of prototypes, quasi models, etc., or it may involve only simpler ideas of redundancy elimination, removal of extraneous variables, etc. In either case questions can arise concerning the properties of these models when further analyses are to be conducted via parameterizations, duality, etc. These topics are examined in the general context of direct and dual linear programming problems with special reference to boundedness properties of the associated solution sets. It is shown that a bounded solution set in one problem implies an unbounded solution set in the dual problem, unless both are one-point sets. The ideas of projection equivalence are then developed to suggest a possible route for utilizing these one-point solution properties for analyzing or solving linear programming problems. These possibilities might prove useful when, for example, it is desired to simplify an initial formulation while achieving a solution that has additional properties—e.g., boundedness—that are also considered desirable.

From the 1970s onwards, the OR community in Britain engaged in ongoing debate on the future of the discipline, the product of an emerging “crisis of confidence” engendered in part by the end of the “golden age of western economic growth” and the associated downsizing, or abolition, of practitioner groups in the corporate industrial sector. In addition, reservations were expressed concerning the increasing “mathematization” of academic OR in the context of the established “hard” or “classical” paradigm. In this respect, British operations researchers, aided and abetted by a number of American colleagues (notably Ackoff, Churchman, and Miser), engaged in a fundamental reappraisal of the OR methodological repertoire and its client base. Thus, in Britain, a new phase in the history of OR was inaugurated whereby the “positivist/scientist” approach bequeathed by the wartime pioneers was subject to challenge and qualification. Whilst some elements in the American OR community empathized with the emergent British critique, the response (notwithstanding Ackoff et al.) was, on the whole, relatively muted. This conservative American response provides one part of the rationale for this paper. The key issue here is to compare and contrast the tone and content of the Anglo-American debate on the future of OR after 1970: In simple terms, why did British OR practitioners and academics (especially the latter) respond so vigorously to the post-1970 OR critique in marked contrast to their American counterparts? In explaining the differential response, the paper will emphasize the interplay among an array of political, intellectual, and economic factors.

This article recalls some of George B. Dantzig’s many contributions to operations research and the management sciences in his ninety-year lifetime.