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The multi-index transportation problem is defined and a summary of the algorithm described in an earlier paper is given. This paper discusses the theorems justifying the use of the algorithm and gives a necessary condition for the existence of a solution. It also describes the formulation of three special-structure linear-programming problems as examples of multi-index problems. The three problems discussed are the capacitated Hitchcock problem, transportation where there is more than one method of transport available, and the transformation of one type of multi-index problem into another. A final section describes problems that have restrictions in the form of inequalities. [ABSTRACT FROM AUTHOR]

This paper presents an algorithm for the global maximization of a convex function subject to linear inequality constraints. It is computationally finite and is designed to converge rapidly on problems in which there are few local optima or the global optimum is significantly better than most of the other local optima. [ABSTRACT FROM AUTHOR]

In this paper the theorem of the alternative is used to show the equivalence between transitive majority rule and the problem of finding positive solutions to a certain set of linear inequalities. A system of linear inequalities is derived whose possession of a strictly positive solution for every triple of alternatives is both necessary and sufficient for the transitivity of majority rule, The consequence of this is that we can consider not only restrictions on admissible individual preference patterns but also restrictions on the distribution of votes over these patterns to ensure transitivity. [ABSTRACT FROM AUTHOR]

This paper discusses stationary-point optimality conditions for inequality constrained nonlinear programming problems where the functions involved are continuous but not necessarily differentiable. We obtain generalizations of the well known Fritz John and Kuhn-Tucker necessary conditions. We also discuss the sufficient conditions for optimality where the usual convexity assumption is replaced by a weaker assumption of 'supportability.' [ABSTRACT FROM AUTHOR]

Consider the set of all vectors b such that the system g(x)≤ b has a solution, where g:D→R[supm], D⊆R[supn]. This paper presents necessary and sufficient conditions for this set to be closed, with special results on continuous and convex functions, and gives an application to mathematical programming. [ABSTRACT FROM AUTHOR]

This paper extends the Sequential Unconstrained Minimization Technique for nonlinear programming to include problems where the constraints are a mixture of inequalities and equalities. Theorems indicating the dual nature of the method and the solution of a subproblem are given. Finally, the theoretical basis for a k-point extrapolation procedure is established. [ABSTRACT FROM AUTHOR]

A linear programming problem is said to be stochastic if one or more of the coefficients in the objective function or the system of constraints or resource availabilities is known only by its probability distribution. Various approaches are available in this case, which may be classified into three broad types: 'chance constrained programming', 'two-stage programming under uncertainty' and 'stochastic linear programming'. For problems of 'stochastic linear programming' a distinction is usually made between two related approaches to stochastic programming, the passive and the active approach respectively. In the passive approach to stochastic linear programming the statistical distribution of the optimum value of the objective function is estimated either exactly or approximately by numerical methods and optimum decision rules are based on the different characteristics of the estimated distribution. In the active approach, a new set of decision variables are introduced which indicate the proportions of different resources to be allocated to the various activities. One effect of introducing this set of new decision variables in the active approach is the truncation of the statistical distribution of optimal value of the objective function of the passive approach. In this aspect the active approach is useful in suggesting criteria for changing from one optimum decision rule to another, as the probability distribution of the objective function is specified less and less incompletely for different choices of the set of new decision variables. This paper investigates some mathematical relations between the active and passive approach and derives some inequalities for the case when the elements of the coefficient matrix of the inequalities are random variables. Since an 'approximate solution' of a stochastic linear programming problem is defined usually by replacing each random element by its expected value and then solving the resulting non-stochastic program, some conditions have been derived under which the expected value of the objective function for the optimal solution will exceed the optimal value of the objective function for 'the approximate solution'. Based on the properties of the distribution of extreme values and other order statistics, some approximate bounds have also been specified for the passive approach, which can easily be extended to the active approach. [ABSTRACT FROM AUTHOR]

This paper describes a method of minimizing a strictly convex quadratic functional of several variables constrained by a system of linear inequalities. The method takes advantage of strict convexity by first computing the absolute minimum of the functional. In the event that the values of the variables yielding the absolute minimum do not satisfy the constraints, an equivalent and simplified quadratic problem in the `Lagrange multipliers' is derived. An efficient algorithm is devised for the transformed problem, which leads to the solution in a finite number of applications. A numerical example illustrates the method. [ABSTRACT FROM AUTHOR]

Duality gaps which may occur in semi-infinite programs are shown to be interpretable as a phenomenon of an improper representation of the constraint set, uTPi ≥ ci , i ε I. Thus, any semi-infinite system of linear inequalities has a canonically closed equivalent (with interior points) which has no duality gap. With respect to the original system of inequalities, duality gaps may be closed by adjoining additional linear inequalities to the original system. Also, for consistent, but not necessarily canonically closed programs, a partial regularization of original data removes duality gaps that may occur. In contrast, a new ‘weakly consistent’ duality theorem without duality gap may have a value determined by an inequality which is strictly redundant with respect to the constraint set defined by the total inequality system. [ABSTRACT FROM AUTHOR]

The inequality Pr (a ≤ W[sub i] ≤ b; i = 1, 2, ···, n) ≥ II[sup n, sub i = 1] Pr (a ≤ W[sub i] ≤ b) is shown to hold when the random variables W[sub 1], ···, W[sub n] have a particular multivariate f-distribution which is defined. It is used in this paper to construct conservative simultaneous prediction intervals for failure times in certain life test experiments. An additional inequality is shown to hold for all bivariate distributions belonging to a class of bivariate f-distributions which is defined. An indication of the use of this latter inequality is given. Some tables are included which indicate the sharpness of each of the inequalities. [ABSTRACT FROM AUTHOR]

The Kuhn-Tucker necessary conditions for a local maximum of a differentiable function subject to differentiable inequality constraints, are derived by showing that the classical necessary conditions for a local maximum of a differentiable function of N variables subject to less than N equality constraints, are also meaningful when there are N or more equality constraints, if appropriate assumptions, similar to Kuhn and Tucker's `constraint qualification,' are made. [ABSTRACT FROM AUTHOR]

The article describes a method to obtain a local maximum feasible solution of a quadratic function of variables subject to linear inequalities. It shows how the feasible area might be appropriately reduced, so as to ensure that another, and usually higher, local maximum is obtained. In the absence of computational experience, one may be allowed to hazard the guess that the problems encountered when one climbs to a saddle-point, are similar to the problems of degeneracy, in providing more theoretical than practical difficulties. The authors assume that the feasible plan in and the expression of the objective function refer to a feasible local maximum value of the objective function.

The article explores constraints which enter in programming problems in the form of linear inequalities. The solution of a programming problem consists of finding that point out of the points satisfying all constraints which maximizes some given objective function. Constraints may be trivial in the sense that, should we delete them, the set of points satisfying all (remaining) constraints does not increase. This notion is discussed in detail using the concept of a trivial constraint, proves concisely the main theorems of linear programming, known as the duality theorem and the existence theorem. And indeed it may be useful to stress here that the purpose of this note is not to suggest that in practice of constraints should be checked for possible triviality, but rather, that they could conceivably be so checked. And that, therefore, for theoretical purposes it can always and without loss of generality be assumed that there are no trivial constraints in the original formulation of the problem.

It is often rightly stressed that research workers should learn how to sell their work to managers, by avoiding technical jargon, and by otherwise acquiring skill in exposition. But managers often take a step towards the research men by learning a little science. I wish to advocate that managers should learn to use a little of the language of the theory of rationality, to make judgments of inequalities between probabilities and between utilities, and to understand what it means to be reasonably consistent. In my opinion the consequences would be a great increase in the value of operational research. A brief description of what I mean by rationality is followed by a tentative definition of "degrees of rationality". Here it seems to be necessary to make use of the notion of rationality of "types I and II". The remainder of Section 3 is concerned with an elaboration of some aspects of rationality that seem to me to be important, and either somewhat novel or not often enough emphasized in print. In Sections 4 and 5 some of the benefits and difficulties associated with the use of the theory are mentioned. [ABSTRACT FROM AUTHOR]

The article describes a procedure to maximize a strictly concave quadratic function subject to linear constraints in the form of inequalities. First the unconstrained maximum is considered; when certain constraints are violated, maximization takes place subject to each of these in equational (rather than inequality) form. The constraints which are then violated are added in a similar way to the constraints already imposed. It is shown that under certain general conditions this procedure leads to the required optimum in a finite number of steps. The procedure is illustrated by an example while also a directory of computations is given. [ABSTRACT FROM AUTHOR]