ISSN:0028-1441    

DOI:10.1002/nav.3800060102

出版商:Wiley Subscription Services, Inc., A Wiley Company    

年代:1959

数据来源: WILEY

摘要:

AbstractThe purpose of this paper is to show, on an example, how rather forbidding algebra may often be avoided by quite elementary considerations concerning flows in a properly constructed flow chart or network. The problem of production smoothing is incidental in the sense that it could have been replaced by other illustrations of the technique.

ISSN:0028-1441    

DOI:10.1002/nav.3800060103

出版商:Wiley Subscription Services, Inc., A Wiley Company    

年代:1959

数据来源: WILEY

摘要:

AbstractIn the Naval Research Logistics Quarterly of March 1958, the author published some exploratory research on ‚Loss Resulting from the Use of Incorrect Data in Computing an Optimal Inventory Policy.”︁ The present paper continues these investigations by examining the effect of errors in the loss function (Part I) and in the discount rate (Par

ISSN:0028-1441    

DOI:10.1002/nav.3800060104

出版商:Wiley Subscription Services, Inc., A Wiley Company    

年代:1959

数据来源: WILEY

摘要:

AbstractThe procedure which involves successively minimizing a convex function with respect to each of its coordinates is shown to converge if constraints are rectangular and the function has continuous derivatives. It is also shown that certain more general procedures, which might be expected to converge more rapidly, are convergent. A criterion is exhibited which yields information concerning the rate of convergence of this method.

ISSN:0028-1441    

DOI:10.1002/nav.3800060105

出版商:Wiley Subscription Services, Inc., A Wiley Company    

年代:1959

数据来源: WILEY

摘要:

AbstractAn algorithm is presented for solving the transportation problem when the shipping cost over each route is a convex function of the number of units shipped by this route.The algorithm can also be used when the shipping costs are linear, whether or not there are capacity restrictions on the number of units that may be shipped over each route. But it does not skem to be any better than other standard methods for these problems, such as those given by Ford and Fulkerson [6], [7].

ISSN:0028-1441    

DOI:10.1002/nav.3800060106

出版商:Wiley Subscription Services, Inc., A Wiley Company    

年代:1959

数据来源: WILEY

摘要:

AbstractHistorical. The topic of games with vector payoffs is one which could be expected to attract attention on the basis of its intrinsic interest. However, the history of the particular problem treated in Dr. Shapley's paper was not of this kind and m a y have some interest of its own. During an interval when the writer of this note was engaged in operations research his group was asked t o analyze a combat situation in which movement of f o r c e s and the inhibition of such movement were critical It turned out to be feasible to represent essential aspects of this situation by a game‐like model with reasonably well defined courses of action corresponding to pure strategies for each side. However, each pair of these strategies, one for each player, generated both a time delay in the movement and losses to the moving forces not balanced by losses to the inhibiting player. That is the model was a game with vector pay off in the sense of this paper. Since the values of these delays and losses would be realized in a subsequent battle in the area approached by the movement in question, it was clear that some s o r t of exchange ratio between attrition and delay must exist. Efforts to obtain estimates of such a ratio failed completely. In fact, several rather arbitrary weightings of delay and attrition were assumed and the resulting numerical games solved for the sake of the insight they could provide, but this was a rather unsatisfactory expedient. Had the theory presented in this paper been available, it would have been possible at least to sharpen the questions asked in pursuit of the „exchange ratio,”︁ and probably to eliminate rationally considerable sets of strategy pairs a s unsuitable or a s representing extreme cases in the weighting.In any event, the theory was not a t hand and the study could not wait on its development. The problem was, however, brought to the attention of Dr. Shapley, with this paper a s a consequence.Nature of the Results. Perhaps a comment on the kind of results which are presented is in order since there is a sort of fitness to them. As was noted above, the occurrence of vector payoff represents a failure to resolve some of the questions whose answers are needed in order to construct a game model. This being the case, one should not be disappointed that the theory does not produce a clear cut, well defined solution concept. What it does produce, if the payoff vectors have K + 1 components, is a ZK parameter set of equilibrium points, in comparison with a unique solution for a numerical game similarly specified. (The fact that numerical game solutions need not be unique has its counterpart among vector game equilibrium points also, of course.) Such an „equilibrium set,”︁ however, need not cover a major part of the joint mixed strategy space, even in games with only a few pure strategies for each player. Accordingly, although finding these sets may be extremely tedious for a particular game, they may well turn out to be worth having in the sense that they limit the domain over which weighting comparisons of the payoff vector components need be made. Since Dr. Shapley's results concern both „strong”︁ and „weak”︁ equilibrium points it is pertinent to note that this distinction appears „usually”︁ to concern the boundaries of the „equilibrium sets.”︁ That is one might expect that the set of strong equilibrium points omits boundary points which are included in the weak case. It should be remarked, however, that this characterization of the distinction is not precise or universal, and ins

ISSN:0028-1441    

DOI:10.1002/nav.3800060107

出版商:Wiley Subscription Services, Inc., A Wiley Company    

年代:1959

数据来源: WILEY

ISSN:0028-1441    

DOI:10.1002/nav.3800060108

出版商:Wiley Subscription Services, Inc., A Wiley Company    

年代:1959

数据来源: WILEY

摘要:

AbstractA recent result of Gomory (Princeton) solved an outstanding problem, namely that of solving linear programs in integers. Gomory showed how to add linear inequality constraints to a linear programming problem automatically in such a way that the extreme points of the resulting convex contain only integral solutions in the neighborhood of the minimum. In this paper an alternative method is given for generating these additional constraints in a way easy to justify and to apply. However it is not known whether these conditions will lead to a solution in a finite number of iterations of the simplex method as is true for the stronger Gomory conditions. Thus anyone considering their practical use should weigh the ease of generation against the extra number of iterations that will undoubtedly be required for convergence.

ISSN:0028-1441    

DOI:10.1002/nav.3800060109

出版商:Wiley Subscription Services, Inc., A Wiley Company    

年代:1959

数据来源: WILEY

摘要:

AbstractLinear programming has many worthwhile applications to transportation problems of a relatively uncomplicated nature. For inore complicated problems, however, a realistic linear‐programming formulation can require such an excessive number of variables that obtaining a solution is not feasible. This paper shows that an excessive number of variables is usually required when different types of items are to be transported subject to time restrictions and on carriers that can simultaneously handle several item

ISSN:0028-1441    

DOI:10.1002/nav.3800060110

出版商:Wiley Subscription Services, Inc., A Wiley Company    

年代:1959

数据来源: WILEY

ISSN:0028-1441    

DOI:10.1002/nav.3800060111

出版商:Wiley Subscription Services, Inc., A Wiley Company    

年代:1959

数据来源: WILEY