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In a paper on ‘Stationary Properties of a Two-Echelon Inventory Model for Low-Demand Items’ [Opns. Res. 19, 761–773 (1971)], R. M. Simon presents an argument for finding the stationary distribution of inventory position that is not always valid. This note gives a corrected version of Simon's method. [ABSTRACT FROM AUTHOR]

This paper extends the well known results for linear fractional programming to the class of programming problems involving the ratio of nonlinear functionals subject to nonlinear constraints, where the constraints are homogeneous of degree one and the functionals are homogeneous of degree one to within a constant. Two rather general auxiliary problems are developed, and the relations between the solutions of the auxiliary problems and the solutions of the original problem are codified. Applications of the results for specific problems are also presented. [ABSTRACT FROM AUTHOR]

This paper deals with the availability and the reliability of a two-unit system with a warm standby and subject to a single repair facility. It assumes the failure times of the units to be exponentially distributed with parameters λ and λ1, respectively, initially, a unit is switched on and the other one is kept as a warm standby. The system breaks down if a unit fails while the other is still under repair. This paper characterizes the system by the probabilities of its being in the up or the down state, sets up integral equations for these probabilities by identifying suitable regenerative stochastic processes, and employs the Laplace-transform technique to solve these equations. It obtains the mean down-time of the system during (0, t) and the mean time to system failure explicitly. [ABSTRACT FROM AUTHOR]

A simple extension of the Lagrange optimization method is valuable in allocating defense resources among a large number of independent defense locations. This kind of allocation problem is complicated by the fact that the attacker can optimize his attack against whatever defense is chosen. This paper describes an extended or double Lagrange method, which provides strictly optimal allocations for the attacker, but not necessarily for the defender. Nevertheless, by careful use of the method, it is possible to obtain defense allocations that are at least approximately optimum. The major part of the paper provides a technique for verifying results and computing bounds for the error—a technique that is necessary because the uncritical application of the method can occasionally give results that are far from optimum. [ABSTRACT FROM AUTHOR]

The article features the author's comments on the research paper "A New Product Analysis and Decision Model." The author acknowledges the important applications of the paper for sequential decision theory in the field of management science, but expresses concern as to generalizations made in the text. It is suggested that the demand function component of the mathematical analysis discussed in the paper. The author proposes that the functional analysis overlooks some main difficulties associated with the featured decision making variables. The industrial product development process is discussed. A heuristic programming approach is also examined.

This paper describes and discusses some search problems, starting with the case of no false contacts, to provide a foundation for developing a search theory in the presence of false contacts. Important properties of the search plan are defined and illustrated by examples. After a general description of false-contact generators and alternative actions that may be used when contacts are made, the problem is formulated for the case in which false contacts are generated by real stationary objects that are investigated when contacted. It is shown that the formulation and solution of the standard optimization functionals are contingent on the number of false targets found, in general, and possibly on the locations and times of contact as well. The optimization functional is difficult to write and more difficult to solve. The formulation is made for the expected-time functional when the number of false targets is bounded. Finally, the paper outlines a procedure for ending the solution and illustrates it with an example. [ABSTRACT FROM AUTHOR]

This paper considers a class of optimization problems characterized by constraints that themselves contain optimization problems. The problems in the constraints can be linear programs, nonlinear programs, or two-sided optimization problems, including certain types of games. The paper presents theory dealing primarily with properties of the relevant functions that result in convex programming problems, and discusses interpretations of this theory. It gives an application with linear programs in the constraints, and discusses computational methods for solving the problems. [ABSTRACT FROM AUTHOR]

This paper defines a problem type, called the transportation-location problem, that can be considered a generalization of the Hitchcock transportation problem in which, in addition to seeking the amounts to be shipped from origins to destinations, it is also necessary to find, at the same time, the optimal locations of these sources with respect to a fixed and known set of destinations. This new problem is characterized mathematically, and exact and approximate methods are presented for its solution. [ABSTRACT FROM AUTHOR]

Even for a stable one-server queue with relative traffic intensity less than unity, the maximum queue length observed will tend to infinity in probability as the length of time for which the system has been observed becomes large. It is important to have information on the behavior of the maximum queue length to enable estimation of the congestion likely to be encountered in a system operating over a long period. The present paper studies the stable GI/M/1 system, and, interpreting the results obtained rather loosely, one can say that the maximum queue length attained by the time the nth customer has arrived (n large) can be expected to be of the order of log n. [ABSTRACT FROM AUTHOR]

Stochastic systems with multiple inputs are common in practice. This paper investigates such a system with random and scheduled inputs, and determines the equilibrium behavior of the basic process (a waiting-time process if the system is a queue, or a storage process if it is a storage system) by reducing the problem to the solution of an integral equation, which is solved by the method of factorization. [ABSTRACT FROM AUTHOR]

This paper studies a particular random tour (continuous random walk): A particle moves in two dimensions at constant speed by choosing successive travel directions that are independent and uniformly distributed between 0 and 2π, with the lengths of the steps between direction changes being independent, exponentially distributed, random variables. An analytic expression for the probability density of the particle's position after a time t is derived, and an application is made to a military situation where the 'particle' is a target trying to escape being destroyed by an unseen enemy. The bulk of the paper is devoted to deriving the density function. [ABSTRACT FROM AUTHOR]

This paper describes a two-person zero-sum game in function space concerning the optimal allocation of a homogenous defense force on different levels of readiness. The second player, the attacker, wants to determine the time for the attack such that the cost is minimized. As the defense is subject to attrition, a delay is preferable for the attacker. This is balanced, however, by a delay cost which increases with time. The existence of a solution consisting of a pure defense strategy and a mixed strategy for the attack time is proved elsewhere. This paper mainly treats an approximation of the game with a time-discrete version for which an algorithm is derived. Arbitrarily close upper and lower bounds of the value of the original game can be obtained by the algorithm, as well as feasible defense and attacker strategies that realize these value estimates. Computational results are given for two simple problems, one of which is also solved analytically. [ABSTRACT FROM AUTHOR]

This paper shows that the Lagrange multiplier method suggested by H. Everett does not have 'Fail-Safe' properties when applied to simple situations with nonconvexity only in the functional. Although each Lagrange multiplier problem for these situations has immediate solutions no means of combining them to obtain a solution to any reasonable approximation appears to be visible. [ABSTRACT FROM AUTHOR]

For a service facility serving more than one class of customers a problem arises as to the order in which customers should be served (i.e., the priority rule). A typical example is a manufacturing facility that produces two products to meet a random stream of incoming orders. In this paper the alternating priority nile (also known as the `zero switch rule') is introduced and investigated. Two classes of customers are served by a single service facility. Customers of class i(i= 1, 2) have priority over customers of class j(j=I, 2; ≠1) whenever a class i customer is in service. When the service facility is idle, the first arriving customer enters service and acquires the priority right for customers of his class. Within classes the `first-come, first-served' rule is observed. Customers' arrivals are assumed to be Poisson and service times are assumed to be arbitrarily distributed in- dependent random variables. The steady-state densities of queuing times are formulated by the use of a special mathematical procedure. The expectations of queuing times and sizes of queues as well as the first two moments of the busy periods are obtained in terms of the basic parameters of the arrival process and service time distributions. All the results are related to the case where switching from one type of customers to another is not penalized by setup time. The alternating priority rule is compared to the 'head-of-the-line' and the 'first-come, first-served' rules with respect to expected queuing times and queue sizes. The last part of the paper discusses the possibilities of extending the suggested rule to cases involving more than two classes of customers and nonzero setup times and setup costs. [ABSTRACT FROM AUTHOR]

The concept of social roles with special reference to sex and age roles has been the subject of increasing sociological interest. But the roles in various segments of the society requires further systematic empirical study. This paper attempts to outline what is believed to constitute a fruitful theoretical orientation for such research and to illustrate the application of this theoretical approach in a pilot study, involving twenty intensive case histories of middle class urban married women, in the summer of 1949. It was focused on the "problem" aspects of sex roles and while the discussion will be thus delimited, the theoretical approach it advocates appears equally applicable to other aspects of this general subject. Thus, the current article deals with three main objectives: the functional significance of sex roles becomes apparent; cultural contradictions can be located and finally, possibilities for change can be assessed. Furthermore, though to an extent shared by others, it is the central concern of this orientation to explore the unintended consequences of relevant social and cultural systems, not only for personality but for other systems also.

The smoothing of fluctuations in production and inventory has been handled successfully as a problem in servomechanisms: an optimal linear filter (production scheduling policy) balances costs of inventory fluctuation and costs of production fluctuation. This approach requires the system to respond (by making production rate changes) either continuously or at regular intervals: the usual assumption is that the cost of responding is proportional to the size of the response. This paper explores some reasons why the frequency of production changes may have to be controlled, and offers a new class of policies, called random walk policies, to accomplish this. One of these, the (a, b, c) policy, is investigated for stationary operating characteristics, and a specific example is provided to illustrate the technique of finding optimal values for policy parameters. Some of the characteristics of systems in which random walk inventory policies may be useful are discussed; and more general techniques for identifying the form of the optimal control policy are mentioned. [ABSTRACT FROM AUTHOR]

This paper reviews ideas of American geographers on geographic land form analysis and proposes a framework within which geographic land form studies can be identified and related to other land form studies. The following approaches are identified as geographic 1) descriptive analysis of the present land form, 2) functional analysis of land form, 3) analytical studies of the covariation of individual land form elements; and 4) analytical studies of spatial variations in land form. Selected methods used in such studies are summarized and evaluated. Special emphasis is placed or descriptive methods. [ABSTRACT FROM AUTHOR]

This paper shows that seemingly different 'adjustment' procedures are equivalent, if viewed in appropriate coordinate systems. It extends previous results concerning sequential adjustments that are constrained to be linear functions of observed impact points to the class of translation-invariant procedures. It also reviews properties of the optimal sequential adjustment procedure, including some related to stochastic approximation. [ABSTRACT FROM AUTHOR]

This paper describes a short method of computing the intrinsic growth rate which yields more accurate estimates than other methods which have similar information requirements. The procedure is recommended when hand or desk calculation of the intrinsic growth rate is necessary. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to describe a new gradient method for maximizing general functions. After a brief discussion of various known gradient methods the mathematical foundation is laid for the new algorithm which rests on maximizing a quadratic approximation to the function on a suitably chosen spherical region. The method requires no assumptions about the concavity of the function to be maximized and automatically modifies the step size in the light of the success of the quadratic approximation to the function. The paper further discusses some practical problems of implementing the algorithm and presents recent computational experience with it. [ABSTRACT FROM AUTHOR]

This paper presents the basics of digital logic operations, discusses the functional operation of electronic circuits which accomplish these operations and analyzes the operation of complexes of these circuits which may be used for counting, timing, and amplitude registration of the phenomena of interest in psycho-physiological research. [ABSTRACT FROM AUTHOR]

This paper formulates a convex programming model allocating submarine-launched ballistic missiles (SLBMs) to launch areas and providing simultaneously an optimal targeting pattern against a specified set of bomber bases. Flight times of missiles from launch areas to bases vary and targets decrease in value over time. A nonseparable concave objective function is given for expected destruction of bombers. An example is presented. [ABSTRACT FROM AUTHOR]

This paper is concerned with determining the various probability generating functions and moments associated with the number of tasks performed by a machine subject to breakdown. The running time, waiting time before repair, and the repair time are independent random variables with general probability density functions. The generating function of the number of tasks for a given uninterrupted running time is assumed to be known. Some special cases are considered. [ABSTRACT FROM AUTHOR]

The estimate of expected critical-path length in PERT networks in this paper is usually better than the FULKERSON estimate because of greeter flexibility in choosing a path whose length approximates the length of the critical path. As the calculations are based on a model approximating the actual network, the new estimate may fall on either side of the expected critical path, even though the Fulkerson estimate never exceeded the expected critical-path length. A Fortran program was written to calculate the new estimate, whose calculation is more complicated than Fulkerson's estimate. The new estimate, Fulkerson's estimate, and the expected critical-path length are compared in several examples. [ABSTRACT FROM AUTHOR]

Efficient search of items in a store of information, such as books in a library, abstracts of articles, information in a computer data bank or the like, requires that the items most closely connected in content be dose together, in order that a person looking for some particular information may concentrate his search on as small a part of the store as possible. The degree of ‘connectedness‘ between two items i and j can be expressed in terms of a correlation index nij. This paper discusses methods of estimating these indices. Once the correlation indices are known for each pair of items in the store, it is possible to assign a position xi for each item i, along a linear classification scale, such that the larger the nij between items i and j the closer the two items are on the scale. Such a scale indicates the optimal ordering of books on library shelves or of items in any other data collection. It also provides means for optimal subject descriptors for a computer-based information store. A procedure is described whereby the values of the xi's may be determined in terms of the nij's. When the items in the store tend to ‘clump together’ in classes, with much smaller correlation between items of different classes than those between items of the same class, an approximation technique may be used to locate the relative positions of the classes along the scale, as well as the distribution of the x's within each particular class. An example is given of these techniques. [ABSTRACT FROM AUTHOR]

This paper extends the results of Evans and Gould for stability in mathematical programming. In particular, it shows that their conditions apply to functional perturbation, to equality constraints, and to policy stability under certain conditions. Further, it shows that strictly monotonic programs and positively homogeneous programs possess the closure property needed for stability. Finally, some necessary and sufficient conditions are presented for lower and upper semicontinuity of certain point-to-set mappings. [ABSTRACT FROM AUTHOR]

Methods are available for computing properties of the waiting time associated with a variety of single-server queuing models in the equilibrium state. Since these calculations are more difficult in the time-dependent case, this paper develops an approximation procedure applicable to a wide class of single-server queues. Specifically, it considers the moments of the waiting time Wn. of the nth arriving customer of the system GI/G/1 and (given certain conditions) uses an expansion related to the central limit theorem to derive approximations to these moments. The expectation EWn is treated in some detail, particularly for the Erlang system Ek/Em/1, and numerical illustrations are given. The same arguments apply to higher moments, but the calculations become tedious and details are not given. [ABSTRACT FROM AUTHOR]

SUMMARY A medical example suggests the fitting of a linear functional model with replicated observations and inhomogeneous error variances. For a particular error structure relevant to the example being discussed, the paper considers the maximum-likelihood estimation of the parameters in the model (slope, intercept and error variances). The estimators have no simple explicit form and need to be evaluated by iterative methods. In spite of this, simple closed form expressions are obtained for the asymptotic standard errors of the estimators. Some comments are made on alternative error structures which might be relevant in other biological and medical problems similar to the one considered in the paper. [ABSTRACT FROM AUTHOR]

This paper takes a few steps toward alleviating problems of data analysis that arise from the fact that elementary expressions for density and cumulative distribution functions (c.d.f.'s) for most stable distributions are unknown. In section 2 results of Bergstrom [3] are used to develop numerical approximations for the c.d.f.'s and the inverse functions of the c.d.f.'s of symmetric stable distributions. Tables of the c.d.f.'s and their inverse functions are presented for twelve values of the characteristic exponent. In section 3 the usefulness of the numerical c.d.f.'s and their inverse functions in estimating the parameters of stable distributions and testing linear models involving stable variables is discussed. Finally, section 4 presents a Monte Carlo study of truncated means as estimates of location. In every case but the Gaussian, some truncated mean is shown to have smaller sampling dispersion than the full mean. [ABSTRACT FROM AUTHOR]

This paper presents an analysis of second generation "exogamy" in the Kibbutzim in Israel. It combines and coordinates an elucidation of social system functions with a developmental-motivational interpretation. Examination of this case study provides a crucial test of theories of mate selection. In addition it enables us to go beyond the specific problem of mate selection and deal with the more general theoretical problem of the relations between functional and causal analysis. [ABSTRACT FROM AUTHOR]

In this paper a search scheme for finding the maximum of a discrete objective function of several variables is proposed. The scheme is sequential and optimal in the sense that it determines the maximum with minimum number of measurements of the function. Unlike other sequential search techniques with this property the scheme places no condition on the number of possible maxima of the objective function. Instead it requires that some bounds on the rate of change of the function per each variable be known. The points at which measurements of the value of the objective function are to be taken are determined successively by maximization of an auxiliary function updated after each measurement. The algorithm is described in detail, and illustrated by two simple examples. [ABSTRACT FROM AUTHOR]

This paper develops efficient algorithms for obtaining a dual feasible starting basis for the capacitated distribution problem. The motivation for the development of such a technique is threefold. Firstly, although efficient dual methods for solving such problems have been developed, none of these methods have addressed themselves to the problem of efficiently determining a dual feasible starting basis. Secondly, no counterpart method exists for obtaining a primal feasible starting basis for the capacitated problem; furthermore, the development of such a counterpart procedure is unlikely. Finally, because our approach provides a dual feasible basis that is both legitimate (i.e., nonartificial) and good (in terms of the dual functional), the synthesis of our procedure with one of the dual distribution methods should provide a more efficient computational algorithm for solving capacitated distribution problems than primal counterpart methods. [ABSTRACT FROM AUTHOR]

In a preceding paper Jacobsen [3] shows that marginal allocation solves a class of discrete, single constraint, min-max allocation problems. A dual approach to this problem is presented, based on a generalization of Jacobsen's P condition, which should prove more efficient when many marginal improvement iterations would be required. A variant of sequential search by bisection is applied, possibly augmented by an accelerated marginal allocation scheme, in which finite convergence to an optimum solution is guaranteed. [ABSTRACT FROM AUTHOR]

If the constraint matrix of a linear program has special structure it may be possible to speed computation. Techniques have been developed to take advantages of such special structures as generalized networks, generalized upper bounding, and decomposition. For these matrix structures, it is shown in this paper how to extend the techniques to Zangwill's mathematical programming algorithm, the convex simplex method. [ABSTRACT FROM AUTHOR]

The paper presents a new approach, based on analysis and geometrical interpretation, to the solution of Markov stochastic games. The proposed algorithm, using iterations in policy space, turns out to be a Newton-Raphson type procedure. Several numerical examples are given, covering the terminating and non-terminating cases respectively and illustrating the advantages of the proposed algorithm compared with other known algorithms. Special attention is given to Howard's sequential decision problem with discrete and continuous policy spaces. [ABSTRACT FROM AUTHOR]

It is shown that minimax problems with convex-concave payoff and convex inequality constraints can be reduced to solving an equivalent constrained minimization problem. The objective function of the minimization problem is not convex, nor is the set of feasible solutions convex. Nevertheless, it is shown that any point which satisfies the first order necessary conditions for optimality must be a global minimum. [ABSTRACT FROM AUTHOR]

This paper develops a new method for the solution of DEMON-type functional equations. It is shown how the latter can be reduced to solution of a separated system of simpler equations which, for discrete distributions, can be solved by linear programming methods. The reduction also permits general characteristics of the solutions to be inferred. Additionally, methods of approximation and bounding are developed and interpreted for the general case. Because of the importance accorded to log-normal distributions for the new-product marketing problem, further details and developments are undertaken with reference to this class of distributions and then illustrated by reference to the single-parameter case. [ABSTRACT FROM AUTHOR]

The theory presented in Part I of this paper led to a Basic Parametric Procedure for a broad class of strictly concave parametric programs. In this part, additional theory is developed that facilitates efficient computational implementation. An illustrative graphical example is given, and some extensions are indicated. [ABSTRACT FROM AUTHOR]

This paper, which is presented in two parts, develops a computational approach to strictly concave parametric programs of the form: Maximize αƒ1(x) + (1 - α)ƒ2(x) subject to concave inequality constraints for each fixed value of α in the unit interval, where ƒ1 and ƒ2 are strictly concave and certain additional regularity conditions are satisfied. This class of problems subsumes a corresponding class of vector maximum problems and also, by means of a simple device, provides a deformation method for ordinary concave programming. The approach is based on exploiting the continuity properties of the parametric program so as to efficiently maintain a solution to the associated Kuhn-Tucker conditions as α traverses the unit interval. The same approach can be adapted to much more general parametric programs than the one above. In Part I, a Basic Parametric Procedure is derived and shown to be finite in a certain sense. It forms the basis of various parametric programming algorithms, depending on what special assumptions are made on the functions. In Part II, additional theory is developed that facilitates computational implementation, and one possible general-purpose algorithm for a digital computer is given. An illustrative graphical example is presented and several extensions are indicated. [ABSTRACT FROM AUTHOR]

The article discusses work in the field of linear programming. The author attempts to give American readers an idea of the kind of work being done in the field of rational planning in Russia. Specific examples are given, through which the author hopes to convey the types of interpretation that Russians have made of abstract mathematics. The author has made a number of contributions in the field of pure mathematics and also to the theory of functional analysis. Also noted are the author's contributions to applied mathematics in numerical analysis and in the theory and practice of computation.

Brown and Rutemiller have given a procedure for deciding whether or not to continue a fixed-sample test in the face of early failures. This paper presents a method of calculating one of the key quantities of the procedure; it eliminates the need for numerical-integration techniques in many applications and can be used if tables of the binomial distribution or tables of the incomplete beta function are available. [ABSTRACT FROM AUTHOR]

Researcher Francesca Cancian's paper on "Functional Analysis of Change," published in the December 1960 issue of the periodical "American Sociological Review," proposes that "functional analysis" be identified with a certain logical structure or judgment of causal argument. The logical structure is that some "state G" can be maintained by either of two causal variables, and that when it is not maintained by one variable then it tends to be maintained by another. The article questions this logic of judgment as an exercise in social theory. To elaborate this argument, it compares substantive sociological results of this logical exercise with results of a work which gets rather short shrift in the paper, that of sociologist A. R. Radcliffe-Brown. Radcliffe-Brown's theory of "G states" is more powerful and based on real conditions under which lineage systems develop. The development of the discipline depends more on the development of such elegant, powerful, and economical theories, which explain a wide range of data, than on fitting these theories into various logical structures.

The paper provides several axiomatizations of the concept of ‘path independence’ as applied to choice functions defined over finite sets. The axioms are discussed in terms of their relationship to ‘rationality’ postulates and their meaning with respect to social choice models. [ABSTRACT FROM AUTHOR]

In many situations, approximation of a set of data by a polygonal curve is more advantageous than approximation by a polynomial. If the join points of the polygonal curve are known, the problem is quite simple. If, however, they are to be chosen in some expeditious fashion, considerable numerical difficulties can arise if the curve-fitting problem is approached directly. In this paper it is shown that dynamic programming offers a simple direct approach to the determination of an optimal fit. [ABSTRACT FROM AUTHOR]

It is desired to set confidence limits for the probability of successful operation at least until time x[sub 0] of a series system of k dissimilar components. The components follow exponential failure laws, and data are available on failure times of components of each type. Exact confidence limits for k=2 have previously been given by Lentner and Buehler. The present paper deals with a large-sample approximation to the exact solution for arbitrary k. [ABSTRACT FROM AUTHOR]

The robustness properties of four estimators of location are studied with respect to eight distribution types. For each type, the probability density function is symmetric about the median and the range of variate is infinite. For the entire class of distributions, the estimator with the highest guaranteed efficiency is the mean of the middle fifty percent of the sample. This study supplements the paper by Crow and Siddiqui (1967). [ABSTRACT FROM AUTHOR]

This paper gives analytic expressions for the moments and recurrence relations for the first four raw moments of a doubly noncentral t-distribution with v degrees of freedom and noncentrality parameters δ and λ. Table I provides numerical values of these moments for v = 2 (I) 20, λ = 2 (2) 8 (4) 20 and any suitable σ. Two approximations to the t"-distribution, involving the moments, are considered. [ABSTRACT FROM AUTHOR]

This PAPER considers a queuing system consisting of N service channels in series where each channel has an exponential or Erlang holding time and (except for the first channel) a finite queue, and where the input process is such that the first queue is never empty. The measures considered are the steady-state mean output rate and mean number of customers in the system (excluding the first queue). First, a procedure is described, for obtaining these measures, that is relatively efficient computationally. Second, an exceptionally efficient procedure is developed for approximating the mean output rate for the case of exponential holding times. It is demonstrated that this procedure provides an excellent approximation for most cases and that it is computationally feasible for large problems. Third, extensive new numerical results are obtained. [ABSTRACT FROM AUTHOR]