It is pointed out that a significant aspect of the transportation problem is to study the sensitivity of the optimal shipping schedules to variations in the configuration of supplies and demands. The authors attempt to show experimentally that recent developments in the field of artificial neural networks hold promise for this area. The application discussed envisages treating a vector of supplies and demands as a stimulus vector, with the corresponding response vector being the optimal shipping schedule and objective function value. For a suitable set of training cases, the matricesRandSare produced, with each training case involving the solution of a transportation problem, for which SPEAKEASY (a standard software package) contains a conventional algorithm. The research objective is to use training information to define an associative memory matrix that will estimate response (solution) vectors associated with new stimulus (formulation) vectors

A probability-free multicriteria approach is presented to the problem of filtering and smoothing when prior beliefs concerning dynamics and measurements take an approximately linear form. In particular, model discrepancy terms are treated as model specification errors that may not have any meaningful probabilistic description. Applications are envisioned in various fields, particularly in the social and biological sciences, where obtaining agreement among researchers regarding probability relations for discrepancy terms is difficult. The essence of the proposed flexible least squares (FLS) procedure is the cost-efficient frontier. This frontier, a curve in a two-dimensional cost plane, provides an explicit and systematic way to determine the efficient trade-offs between the separate costs incurred for dynamic and measurement specification errors. The estimated state sequences whose associated cost vectors attain the cost-efficient frontier, referred to as FLS estimates, show how the state vector could have evolved over time in a manner minimally incompatible with the prior dynamic and measurement specifications. A Fortran program, GFLS, for implementing the FLS filtering and smoothing procedure for approximately linear systems is provided.>

In many engineering problems we must find the roots of an equation of the form\det B(\lambda)=0where B(λ) is a nonlinear matrix of λ. In this paper we show how this problem may be reduced to integrating a system of ordinary differential equations subject to known initial conditions. The method covers also the case of complex roots. It gives directly the roots i: without the need of forming and solving the characteristic equation.

Utilizing the theory of invariant imbedding, a Cauchy system suitable for real-time calculation is presented for the solution of the basic Fredholm integral equation of linear least squares filtering theory when the signal process is stationary. Extensions are also mentioned for more general nonstationary processes.

The Fredholm integral equation where the kernel is semidegenerate has many applications. The solution of this integral equation may be studied as a function of the upper limit of integration x , while t remains fixed. It is shown that the solution satisfies an initial-value problem. This reformulation is well suited to numerical solution by analog and digital computers. The present paper is one of a series on initial-value methods for Fredholm integral equations. Its considerations are of practical significance since an arbitrary kernel may be approximated by a degenerate kernel to a desired degree of accuracy using standard techniques. Furthermore, the important cases in which the kernel is a Green's function and in which the integral equation is a Volterra equation are both covered by this treatment.

In this paper we wish to initiate the study of the application of dynamic programming to the domain of problems arising in the synthesis of logical systems. In a number of fields one encounters the problem of converting a system in one state into another state in a most efficient fashion—in mathematical economics, in the theory of control processes, in network theory, and in trajectory processes. Here we wish to consider a type of question which arises in the design of computers and switching circuits. We shall first treat the problem in general terms, and then consider a special example to illustrate the methods.

By imbedding a complex physical process under consideration within an appropriate class of processes and expressing the functional relationships among the members of the class, it is frequently possible to obtain insights into the structure of the original process which would not be feasible through consideration of that process alone. Not only may analytical expressions be obtained, but frequently computational tools are forged which make possible the exploitation of modern digital computing machines. By way of illustration, this paper is devoted to a discussion of recent applications o f the functional equation techniques of dynamic programming and invariant imbedding to the study of some problems arising in the theory of adaptive control processes and that of transmission through random media. Still other applications which have been made in modulation theory, communication theory, and network analysis are briefly sketched.

In this paper we wish to show that the fundamental problem of determining the utility of a communication channel in conveying information can be interpreted as a problem within the framework of multistage decision processes of stochastic type, and as such may be treated by means of the theory of dynamic programming. We shall begin by formulating some aspects of the general problem in terms of multistage decision processes, with brief descriptions of stochastic allocation processes and learning processes. Following this, as a simple example of the applicability of the techniques of dynamic programming, we shall discuss in detail a problem posed recently by Kelly. In this paper, it is shown by Kelly that under certain conditions, the rate of transmission, as defined by Shannon, can be obtained from a certain multistage decision process with an economic criterion. Here we shall complete Kelly's analysis in some essential points, using functional equation techniques, and considerably extend his results.