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This paper will review and generalize the work of Gearing, Swart and Var dealing with the development of a mathematical model to aid the government of Turkey in determining the “best” allocation of the capital budget for tourism among a large group of competing proposals. One of the more unique aspects of this work was the development of a measure of benefit for particular allocation plans which allowed for subjective information to be an integral part of the investment planning model. The computational procedures developed heretofore to derive investment strategies from the model have been based on dynamic programming, integer programming or combined dynamic programming-integer programming approaches. This paper will show how various methods based on linear programming yield satisfactory answers to many questions regarding development policies. The primary advantage of these linear programming methods is that they can be implemented with standard linear programming computer packages which are generally available, and hence eliminate the high cost of developing special purpose computer codes.

This paper presents a multigoal linear programming model designed to analyse the impact of different strategic objectives in the planning of hospitals. The paper argues that it is necessary to adopt an overall point-of-view in planning; that it is necessary to consider the interactions between the hospital, other social and health institutions and the community at large in order to attain an efficient plan. A simplified cost-benefit analysis is suggested, and it is demonstrated that the opportunity costs of lost working days may be more important than both capital and operating costs from a social point of view.

The automatic plastic design of structural frames can be treated by the method of linear programming. The number of variables, however, increases so fast with the complexity of the frame that only simple frames can be handled by this method even on a large electronic computer. In the present paper a method is proposed, which considers alternately two different requirements that a frame must satisfy, and thereby greatly reduces the size of the problem. Following the Introduction, Part I of the paper presents the method with reference to a simple numerical example; Part II establishes the general applicability of the proposed method.Part III presents some lemmas of practical importance, and some discussion, with examples, of special considerations that may arise in the design of actual frames.

ESTIMATES are established below for the closeness of the optimal values and optimal solutions of the problems (original) and , where M0 and M are convex polyhedral sets of the space Rn, given by sets of linear inequalities, L(x) is a linear function defined on Rn, K is a vector parameter, and ϕ(x, M0, K) is a function chosen by some method or other, representing a measure of the penalty for violating the restrictions defining the set M0. The reason for considering the connection between these problems is that for certain constructions of the function ϕ these problems are either asymptotically equivalent, or are simply equivalent (the latter indicates that their optimal values and optimal sets coincide). The idea of the penalty method is natural and has also been used previously in some form in some papers. It appears, for example, in the procedure of the direct differential method [l]. Methods for the asymptotic reduction of convex programming problems to unconditional extremum problems (or the corresponding numerical realizations) [2, 3] are connected with this idea. The utilization of the penalty method for solving transport problems with prohibitions is given in [4]. It must be mentioned that as distinct from methods of asymptotic reduction of extremum problems with restrictions to unconditional problems, here in the construction of the function ϕ the possibility is foreseen of taking account of the arbitrary part of the restrictions of the original system, and the set of solutions of the remaining subsystem is optimized. This determines the possibility of the applications of some special algorithms for solving optimization problems with “almost” special sets of restrictions or even with an arbitrary set of restrictions. For example, let a linear programming problem be given in the form: find max L(x) if x ϵ {x| (cj, x) - αj ⩽ 0 (j ϵ I1), (cj, x) - αj = 0 (j ϵ = I2), x ⩾ 0}. Putting we (as will be shown below) asymptotically reduce the original problem to the problem max [L(x) - ϕ] subject to the restrictions x ⩾ 0, zj ⩾ 0 (j ϵ I1). But the latter is a problem on the maximization of a concave quadratic function with the reouirement of non-negativeness of the variables, the solution of which can be effectively realized, for example, by the method of gradient projection [5]. In Section 2 of this paper the algorithm corresponding to this case is described, and in Section 3 the results of numerical experiments are discussed.

It is the purpose of this paper to bring to the attention of the chemical engineering profession a recently developed mathematical technique, called “linear programming”, which is exceedingly valuable in solving certain general problems connected with, inter alia, the efficient design of a chemical plant, and the discovery of an optimum production schedule which would maximize the profits of an enterprise. The general and more important features of linear programming will be discussed in the main body of this paper, while, in the Appendix, an illustrative problem of, it is hoped, general interest will be worked out numerically.

The purpose of this paper is to present a rational and systematic method from which state resource allocation strategies to local school districts in state educational finance programs may be determined. In order to determine “optional” resource allocation strategies, a linear programming model was developed for a foundation type state support program. The model, presented in this paper could be solved under various objective functions and constrained by different “design criteria” related to the variances in per pupil expenditures for those districts on the program. The results obtained by the application of the model to an actual foundation program indicate potential usefulness of this approach to school finance. The important contribution of this approach is that it realistically places the problem of educational finance in a system constrained by political, social and economic considerations.

This paper treats continuous-linear programming problems with time delays in the constraints, which are expressed in the form of difference-differential inequalities. Lagrange multiplier functions are used to adjoin the constraints to the cost functional and to compute, by explicit algorithm, a candidate solution. It is proved that if this solution is feasible, then it is also optimal. A similar method is developed for dealing with dual problems. When both the primal and the dual problems have solutions, the relationship between their solutions is exhibited and it is shown how, in some cases, solutions to the dual problem can be constructed from the solutions to the primal problem and vice versa. Finally, the differences between continuous linear programming problems and problems of classical calculus of variations and modern optimal control are discussed. The main advantage of the method proposed in this paper, for computing trial solutions, is that it is quite straightforward and simple and that it does not depend on the simultaneous existence of solutions to both the primal and dual problems.

The paper is concerned with beams and frames with segmentwise constant cross-sections. Chan3 has recently given an optimality criterion for a structure of this kind when neither one of two alternative loadings is to exceed the plastic load carrying capacity. In the present paper, Chan's results, which were derived by means of linear programming theory, are rederived by purely structural arguments.

In this paper the efficiency of the allocation of Czech trade with the member countries of OECD is investigated by means of a linear programming model. We find that considerable losses in this trade occurred in both of the years 1962 and 1964 examined. The bulk of the losses occurs in the allocation of exports and their magnitude is very dependent on the foreign elasticity of demand. Balance of payments constraints, on the other hand, are not an important cause of misallocation.

The paper reviews the early work of Herbert-Stevens in which linear programming was used to find a competitive equilibrium to an urban land market. First, it is demonstrated that a solution to the Herbert-Stevens model does not meet well-established criteria for an equilibrium. Secondly, a new linear programming model is suggested which is proven to achieve equilibrium if certain conditions are met. An iterative procedure for meeting these conditions is suggested, and an operational version of the model exhibits no problem in obtaining convergence. The revised model represents a feasible way of simulating urban land markets.

Two and three dimensional structures are dealt with, subjected to variable repeated loads, in order to establish a numerical tool for determining the load domain multiplier that gives rise to shakedown. The structure is made discrete by finite elements and the yield domain is linearized. By applying Bleich and Melan's theorem, two primal static formulations are found in linear programming, from which the relevant dual kinematic versions are obtained via duality properties. Numerical results are given at the end of the paper, together with some considerations about the numerical efficiency of the proposed formulations.

Most models of urban housing markets are of the simulation (descriptive) variety, apparently because these markets were felt to be so imperfect as to preclude extremal behavior, hence the use of analytically-rich optimization techniques. In this paper, housing markets are split into two parts: an allocation subsystem which distributes housing to households and a construction subsystem which distributes resources to construction in various sub-markets. It is shown that when certain restrictions on prices are satisfied, the allocation subsystem can be characterized as operating on the extremal principle of maximizing aggregate revenue to housing suppliers. The allocation subsystem is modelled using transportation techniques and the dual variables are used as the interface between the allocation subsystem and the construction subsystem. It is assumed that, after allowing for start-up lags, communication and competition are sufficiently great for the construction subsystem to operate in an optimal manner also, and it is modelled via linear programming. The overall system is made dynamic by using sequential processing. An illustration of the allocation model is given and policy implications of the model are discussed.

This paper briefly describes a linear programming model designed to allow the exploration of questions surrounding the management of the environmental impacts of integrated iron and steel facilities. In particular, the model can show how plant discharges will change in the absence of specific legal restrictions or effluent charges, with such variables as product mix, steel-furnace type, casting technology, and the scrap-ore price ratio. In addition, the costs implied by placing restrictions on discharges of specific residuals (e.g., BOD, oil, suspended solids, particulates) may be estimated, or response to proposed effluent charges may be predicted.

Mr. Holloway joined the Gulf Oil Corporation in 1939. His previous publications are in the fields of chemical engineering, statistics and computing. He is now the Senior Adviser on Planning Systems and member of Gulf's European Planning Council. The paper presents the author's view that linear programming is an efficient vehicle for long range planning, that it is more comprehensive and when properly structured requires no more computer time than the perhaps more conventional simulation models. Further, that it has the great advantage that the planner himself can control not only the data content of his model, but can also control the way in which the data is treated within the model even to the extent of making drastic matrix changes.

The paper describes the formulation of the general planning model, TOPAZ (Technique for the Optimum Placement of Activities into Zones) and its application to evaluating and optimizing alternative growth patterns in urban systems. TOPAZ identifies an urban system as a set of interacting activities to be allocated to a set of zones to maximize an objective of overall benefit less cost of interaction between activities together with the benefit less cost of establishment of the activities, over a set of time periods. The model takes the form of a non-linear assignment problem with linear constraints, and is solved using iterative linear programming. The model is illustrated by application at a macro level to the evaluation of alternative corridor growth patterns for the city of Melbourne (population 2.4 million) over three time periods, 1970–80–90–2000. Two activities are involved, residential and employment activity, and the interactions consist of work, residential, industrial and commercial trips. The city is divided into forty zones.

In this note we describe a method for finding the optimal travelling salesman strategy by means of iterations instead of by solving recurrence relations [1, 2]. This method enables us to take into account the possibility of sending out more than one, say l, salesmen at once. As far as computer store is concerned the method is equivalent to the dynamic programming method when l = 1. The logic of the iteration process for finding the optimal travelling salesman strategy does not require a study of the complicated problem of integral linear programming [3]. In this paper the problem is formulated for the case of l salesmen.

The application of quadratic programming to the design of digitally compensated control systems is described and illustrated with three examples. Previous papers (1, 2) discuss in detail the application of linear programming to the design of digitally compensated control systems with deterministic inputs and give a number of examples. Here the application of the more general and more powerful method of quadratic programming (3) is presented. Comparisons are made with the results obtained by linear programming design.

For some years research on solution theorems in probabilistic programming has been dormant. The obvious consequences of formal similarities to deterministic mathematical programming problems had been rapidly exhausted by researchers. Currently, however, the deeper study which was taking place during the 'dormant period' has begun to produce results. On the one hand theorems characterizing optimal classes of stochastic decision rules for various general change-constrained problems have been obtained. On the other hand, a great amount of effort has been expended on the special class of problems called linear programming problems under uncertainty, usually 2-stage and under still more special assumptions. The general objective of these specializations has been to attain results and thereby to gain insight and technique to reapproach more fruitfully the more important and general but more recondite probabilistic programming problems. To this end, few abstractions or devices, from finite-dimensional Banach spaces to the Kakutani fixed-point theorem appear to have gone untried, except, perhaps, the ancient one of study and correlation of the existent results of other researchers. It is the purpose of this paper to provide some such correlation and a redirection so that these simpler probabilistic programming problems may be overcome in all generality with new, simpler methods which offer some promise of extension to the more involved chance constrained (and other probabilistic) models.

Choice among risky investments has been described using a chance constrained programming model with a finite number of states of nature. This paper presents a simple combinational algorithm for solving this model which, at worst, requires solving a number of linear programming problems.

IN this paper two versions of a canonical algorithm for discovering all the optimal solutions of a linear programming problem with the condition of non-negativeness of the variables are presented: the first for the case of canonical notation, the second for the standard notation. The algorithm is obtained by applying the results of [1] (respectively [2]) to the following problem which is obviously equivalent to the linear programming problem indicated: for a finite system of linear equations (inequalities), one of the free terms of which is a parameter, to find the greatest value of this parameter for which the system is consistent, and all its non-negative solutions for this greatest value of the parameter.

The recent Supreme Court decision barring “freedom of choice” pupil assignment plans has forced a large number of school districts to devise alternative approaches to school desegregation. One possibility is establishing attendance boundaries to mix races and linear programming is one method of planning such boundary lines. This paper discusses some technical aspects of the linear programming approach, including the formulation of the problem, some alternative constraint possibilities, branch and bound techniques to avoid splitting neighborhoods, and improvement in execution speed.

This paper explores the potential application of mathematical programming in the evaluation of alternative urban plans and improvement programs. After a brief review of past modeling efforts of this type, an examination of the problems likely to be encountered in identifying and measuring specific objectives and programs is presented. The Model Cities Program is used as an example. A simple linear programming model is subsequently developed, built around a matrix of relative effectiveness coefficients, a set of performance standards, and appropriate program budgets. Supporting analytic techniques and information systems also are discussed. Finally, potential applications of the model are surveyed, and examples of its use in testing and evaluating basic Model Cities programs and objectives presented. Examples of the linear programming model are programmed and run for a hypothetical model neighborhood, using real and assumed data from Chicago. The model allocates dollars among various Model Cities program alternatives, either minimizing the total budget spent or maximizing the achievement of specific Model Cities objectives. Objectives are expressed in terms of standardized percentage impacts upon various socio-economic conditions within a model neighborhood. The purpose of the model is to show how a specific operations research technique might be applied to systematically improve our plan evaluation methodologies. Emphasis is placed upon properly structuring the Model Cities evaluation context, rather than upon refining the mathematical technique itself. Much work remains to be done before viable urban improvement programming models can actually be developed. This exploratory study attempts to draw several basic guidelines for continuing research effort. First, the need for consistency and comparability in the selection of quantitative measures or indices of goal achievement is examined. Second, the need to express alternative programs and policies in terms of three essential characteristics—the objectives to which they are related, the impact or effectiveness which they will have in achieving each objective, and the costs associated with varying levels of effectiveness—is illustrated. Third, in order to permit comparisons and trade-offs among different programs and groups of programs, it is essential that a common measure of goal achievement be developed. One way to define such a common measure of goal achievement is to develop a programs-objectives matrix in which each coefficient represent the percentage of objective j which can be achieved by each dollar invested in program i. Fourth, it must be understood that programs-objectives effectiveness matrices of this type will require a prodigious amount of supporting research and analysis in order to predict the individual impacts of alternative programs and policies. Finally the sensitivity analysis of all variables and parameters—standards, budgets, or programs—within a mathematical programming problem can provide a good deal of information on relative costs, effectiveness and trade-offs.

This paper concerns the development of numerical methods for the determination of the ultimate load of two- and three-dimensional structures assuming an ideal rigid-plastic behavior of the material. According to the classical plasticity theory the ultimate load problem can be mathematically formulated as the problem of finding the maximum or minimum of a linear function, whose independent variables are subjected to inequality constraints. In order to use linear programming techniques these generally nonlinear constraints are approximated by sets of linear inequality restrictions. Linear equilibrium or kinematic compatibility equations have then to be formulated, the corresponding coefficients being determined by virtual work methods. This requires the assumption of parametric stress and displacement fields, which are constructed by mean of finite element procedures. Numerical results for two plate-bending models are presented and some possible plate-stretching models are described and discussed.

Structures, in general, are subjected to more complex load programs than those considered in the theory of limit analysis. The shakedown theory presents general theorems allowing to estimate does the structure adapt to the given load program or not. On the basis of previous general results [22] the present paper gives an approximate method of analysis of frames and arches of arbitrary solid cross-section in the case when the influence of axial forces on the stress state cannot be neglected. The problem reduces to the linear programming problem. Numerical results for a portal frame and for a circular arch of rectangular cross-section are presented.

This paper reports some experiments testing the efficacy of linear programming (LP) as a method for predicting flows of commodities between regions of Great Britain. ‘Transportation problem’ solutions were found for the annual movement by road of eleven commodity categories between 78 origin and destination zones in 1964. Road distances were used as a surrogate for freight costs. Data on actual flows of these commodities in that year were available for comparison. For the more homogeneous commodity classes the LP solutions replicated the actual pattern quite well. Another experiment was carried out using the same commodities, on a subset of zones which were more homogeneous with respect to areal extent - namely the 24 major cities in Great Britain: the correspondence between LP solutions and actual flows was much closer than for all 78 zones. A further test involved finding LP solutions for total road plus rail traffic with six commodities which might be described as ‘heavy’ and which use the rail system extensively. These solutions for total flows proved to correlate with actual flows better than solutions for road movements alone. These results would appear to justify continuation of investigations of this kind using a finer commodity - and if possible zonal - breakdown. It is hoped that data on actual flows for 1967/68, which have a more disaggregated commodity classification, will be available soon. The ‘shadow prices’ of the minimizing dual of the transportation problem were produced and examined to see if they made sense in terms of notions of geographical comparative cost advantage variations for production and consumption.

Summary Non-linear programming problems may be met with in the selection of optimal automatic control systems. This paper presents a method of solving a problem in non-linear programming. The essence of the method consists in reducing the set problem to a repeated search for a solution of a linear programming problem and the choice of values for certain additional parameters that are introduced.

A decision-making organization whose output is either 1 or 0 according to which of the numbers, 1 or 0, is predominant in its inputs, is defined as the majority decision element. A Boolean function represented by such an element is called a majority decision function. Elements of this sort play a significant role in the computer field. The present paper deals with various problems concerned with a single element of this sort: that is, in Section 3, the algebraic properties, and a necessary condition for realizability of a given function; in Section 4, the types of functions realizable by a single element; in Section 5, the determination of the structure of the element for a given function by means of linear programming; and in Section 6, properties of the functions of a small number of variables realizable by a single element.

In this paper an algorithm is proposed for solving continuous linear optimal control systems with state space constraints by solving a sequence of linear optimal control systems without state space constraints. The convergence of the algorithm is proved by a method similar to cutting plane algorithm for convex programs in Banach Spaces. It is also shown how to solve the problem by using mathematical programming algorithm on the discretized problem. A numerical example is solved by discretization and mathematical programming.

The state-vector approach is employed to describe discrete control systems with linear plants. A novel formulation is proposed to obtain “minimum-time” solution through the application of linear programming. For a given initial state, the systematic iterative process employed yields the minimum control time required and its corresponding controlling sequence to bring the system to its equilibrium state. The control input is magnitude-limited. This iterative process also reveals whether or not the controlling sequence is unique. When the controlling sequence of the minimum-time solution is not unique, two objective functions are suggested to continue the process, so that a choice can be made to satisfy the minimum fuel criterion. This iterative process is not restricted by the order of the plant. It can be easily executed on a digital computer. An illustrative example is worked out in detail in the paper.

A GRAVITY MODEL FOR TRIP DISTRIBUTION DESCRIBES THE NUMBER OF TRIPS BETWEEN TWO ZONES AS A PRODUCT OF THREE FACTORS; ONE IS ASSOCIATED WITH THE ZONE IN WHICH A TRIP BEGINS, ONE WITH THE ZONE IN WHICH IT ENDS AND THE THIRD WITH THE SEPARATION BETWEEN THE ZONES. THE SEPARATION OR DETERRENCE FACTOR IS USUALLY A DECREASING FUNCTION OF THE GENERALIZED COST OF TRAVELLING BETWEEN THE ZONES, WHERE GENERALIZED COST IS USUALLY SOME COMBINATION OF THE TIME OF TRAVEL, THE DISTANCE TRAVELLED, AND THE ACTUAL MONETARY COSTS. IF THE DETERRENCE FACTOR IS OF THE EXPONENTIAL FORM EXP (-ALPHA C) AND IF THE TOTAL NUMBERS OF ORIGINS AND DESTINATIONS IN EACH ZONE ARE KNOWN, THEN THE RESULTING TRIP MATRIX DEPENDS SOLELY ON ALPHA. IN THIS PAPER IT IS SHOWN THAT AS ALPHA TENDS TO INFINITY, THIS TRIP MATRIX TENDS TO A LIMIT IN WHICH THE TOTAL COST OF TRIPS IS THE LEAST POSSIBLE ALLOWED BY THE GIVEN ORIGIN AND DESTINATION TOTALS. THAT IS TO SAY THE LIMIT IS A COST-MINIMIZING SOLUTION TO THE LINEAR PROGRAMMING TRANSPORTATION PROBLEM HAVING THE SAME ORIGIN AND DESITNATION TOTALS. IF THIS TRANSPORTATION PROBLEM HAS MANY COST-MINIMIZING SOLUTIONS TAKEN IT IS SHOWN THAT THE LIMIT IS ONE PARTICULAR SOLUTION IN WHICH EACH NON-ZERO FLOW FROM AN ORIGIN I TO A DESTINATION J IS OF THE FORM (R SUB I) (S SUB J). A NUMERICAL EXAMPLE IS GIVEN. /AUTHOR/

This paper deals with an optimal signal setting of a fixed-cycle traffic signal based upon a new criterion of minimizing a degree of saturation of a signalized intersection. As preliminaries, a criterion for undersaturation of the signalized intersection is derived as the degree of saturation of the signalized intersection and mean effective green times of streams of lower priorities are determined as linear functions of the green times and the cycle time. It is shown that the criterion of minimizing the degree of saturation of the signalized intersection leads to the well-known rule of thumb for optimal split. Other several results for the optimal signal setting are obtained in case of nonoverlapping phases. The optimal signal setting problem can be solved respectively by linear programming in case of nonoverlapping or overlapping phases and by mixed-integer programming in case of generalized overlapping phases.

Upper and lower bounds to quantum mechanical second order perturbation quantities are found using the method of Linear Programming. In this paper bounds on the polarizability of hydrogen, helium and neon are obtained using the sum rules popularized by Dalgarno.

This paper deals with an optimal signal setting of a fixed-cycle traffic signal based upon the criterion of minimum overall delay. Webster's method which is an approximation method for the optimal signal setting is theoretically elucidated. Furthermore, approximation algorithms which are a refinement of Webster's method and are applicable to all traffic signals are presented. The algorithms are a combination of linear programming or mixed-integer programming and one-dimensional minimization technique.

This paper describes the production planning system developed for the chemicals division of the Boliden Company Ltd., Sweden. The major elements of the production planning system are a linear programming model, a cost model, a scheduling program and a program to forecast the economic outcome. A special organization was established to implement the plans.

It was found by Lloyd Dines that the following two properties of a finite sequence of functions are equivalent: (A) There exists a positive function that is orthogonal to each member of the given sequence. (B) Every linear combination of the given sequence either changes sign or vanishes identically. This paper gives a proof of the above equivalence by a variational method. Thereby the orthogonal function is given explicitly as the positive part of a linear combination of the sequence of functions. Moreover, this proof leads to various generalizations. In particular, it is shown that the following properties are equivalent: (A′) There exists a positive function with nonnegative moments relative to the given sequence of functions. (B′) Every linear combination of the sequence having non-negative coefficients either changes sign or vanishes identically. Next it is shown possible to transform these properties A′ and B′ into statements about dual linear programs A″ and B″. Program A″ concerns the moments of the sequence of functions with respect to a positive function. The objective of Program A″ is to maximize the moment of the first function subject to the constraint that the moments of the remaining functions equal or exceed preassigned values. The dual program B″ is a minimization problem stated in terms of linear combinations of the sequence. From the equivalence theorem it is shown that the maximum of the primal program is equal to the minimum of the dual program. This last theorem on infinite programs is not merely an analogy with the classical duality theorem of finite programming, because by a suitable refinement it is seen to include the classical theorem as a special case. Various other ramifications, counter examples, and applications related to the Dines theory are developed.

The widespread incidence of teacher strikes, usually associated with demands for higher salaries, has in many school districts throughout the country, established the need to develop effective salary evaluation schemes for school district personnel. Presently used methods of determining salary, such as the fixed step salary schedule, are not capable of considering multiple factors in salary evaluation. The purpose of this paper is to formulate, then apply, a linear programming model to determine a school district salary schedule. The model is capable of incorporating all factors which are considered important by teacher unions, school board, etc., in the salary evaluation. In addition, certain union demands or conditions reached as a result of arbitration can be included in the model. Finally limits placed upon school district resources available to support the salary structure can be considered. Essentially, the model can be used by a school district to determine a logical, internally consistent salary schedule which meets specific union demands at minimum school district cost. Finally the use of the model can give school districts the added flexibility of developing salary schedules which permit overlaps in the established salary hierarchies, e.g., it would permit highly qualified teachers to receive higher salaries than low qualified administrations.

This paper describes the work done in evaluating the application of linear programming techniques to on-line cement raw materials blending. The cement making process is briefly described and a generalized blending system is shown. The method of operation of the blending system is outlined with particular reference to the use of the linear program. The choice of linear programming algorithm is discussed and the blending problem is defined. Two different formulations of the problem are set down and an example is given using data taken from a cement plant. Several cases, using both formulations, are run on two different computers each using a different linear programming system. The results are discussed and lead to the selection of the MINMAX formulation for on-line use.

In this paper we develop bounds for the displacement in the solution set of a system of perturbed linear inequalities, then apply these results to find estimates for changes in the solution set of a perturbed linear program.

The present paper develops a decision tool for planning and evaluating public spending programs aimed at improving urban residential neighborhoods. The urban residential frame-work is conceptualized in terms of four community components; housing, education, health and recreation; each component is measured by a level of facilities which grows and decays with time. A directional graph is drawn to show the relationship between the community components, the inputs in the form of public investment and the outputs or social payoffs in the form of improved facilities. The graphic model of the urban residential community is set to a linear programming format for optimizing public investment. Three applications of the model are indicated. They are: (1) determining a minimal cost split for achieving a desired social payoff level; (2) where the minimal cost is not feasible (e.g. public objection) evaluating the impacts of different sub-optimal investment plans which may be required; (3) tracing the effects of external factors on the growth and decay rates of community facilities.

IN many constrained extremal problems the functional to be minimized or one of the functionals specifying the permissible set, is not even Gateaux-differentiable, and has only directional derivatives. We shall describe such problems as unsmooth. Various aspects of the theory of these problems have been dealt with for example, the necessary conditions for a minimum [1–4], existence theorems and strong convergence of the minimizing sequences [5, 6], and correct stipulation of the constraints and stability [7]. Extremely general minimization methods have been developed for the case when the functional to be minimized is differentiable, and the permissible set has a reasonably simple structure (e.g. linear functionals can easily be projected or minimized on it), i.e. for the smooth case ([8], and [9], Sections 5–7). As regards unsmooth problems, all investigations (with the exception of [10]) of minimization methods for then have been under much less general assumptions than those under which the necessary conditions for a minimum have been obtained (mainly for the finite-dimensional case [11–15]or for unsmooth functionals of a special type [16–19]). In [10], an extremely general and computationally simple minimization method in Hilbert space was described. In this method, however, the values of the functional do not necessarily decrease from iteration to iteration, nor do all the points belong to the permissible set, i.e. it is not a monotonic method of permissible directions. This factor complicates its numerical realization, since it is difficult to check the computations. Other general methods applicable to unsmooth problems include the cutting-plane methods devised by Kelley [20] for the finite-dimensional convex case, and extended to infinite-dimensional problems in [9], Section 10. In these methods, however, the successive approximations again do not belong to the permissible set, and in addition, the initial problem is reduced to a sequence of linear programming problems, the number of constraints in which increases with the number of iterations. The present paper describes a monotonic method of permissible directions for a wide class of unsmooth problems in arbitrary Banach space. In its fundamental idea, the algorithm is similar to those in [11–16, 18,19] and represents an extension of them to the case of arbitrary Banach spaces, unsmooth functions of a more general type, and the presence of a number of constraints. The convergence of the method is proved, and its applications to various unsmooth extremal problems (of mathematical programming, theory of games, approximation theory, and optimal control) considered.

The intersection between project oriented evaluation and determination of R&D programs and macroeconomic development planning is characterized as the focus of this paper. The specific difficulties of the planning problem are described and used to derive the criteria for the external validity of the models presented: (1) an adequate representation of the complexity within the goal system, (2) inclusion of direct and indirect benefits attributable to R&D projects, (3) the incorporation of uncertainty, and (4) the incorporation of project and program interdependencies. Two basic types of models are identified, described and evaluated: (1) models attempting to integrate project oriented aspects into macroeconomic structures, (2) project oriented models attempting to include macroeconomic aspects. Linear programming models are presented separately due to their common weaknesses and strengths. The evaluation indicates that those models (a) focusing upon the evaluation rather than the decision, (b) indicating preferred rather than optimal courses of action, and (c) capable of dealing with qualitative elements have the most promising potential. Some recommendations for further research are given.

In this paper the applications of goal programming to industrial and economic planning are reviewed, and it is shown how the linear programming framework can be extended to include a series of goals (or objectives). A planning method is suggested in which the decision maker is presented with a list of all the possible attainable goals from which he picks the most appropriate set.

This paper describes the behavior of a transformer having a plurality of windings mounted on a single core, and discusses applications in the automatic computation of linear algebraic expressions such as simultaneous equations and the like. The computing circuits are superior to prior ones in that they combine the following properties: they contain no amplifiers, or components of limited life such as vacuum tubes; they require no iteration, either manual or automatic; and they bear a close pictorial resemblance with the equations to be solved, thus simplifying the operating procedure. Actual experimental results are given for a set of simultaneous inequalities such as occur in linear programming theory; this includes the solution of several sets of simultaneous equations.

AN iterative method for solving problems of linear programming has been described in [1]. This method is not completely determinate; its basic cycle is determined by two parameters ϵ and K. Its convergence was proved in [2, 3], a sequence e1 > e2 … > eR … → 0; tending monotonically to zero being used for ϵ; as regards K, the number of iterations in an “internal” procedure B, it was assumed in the proof that the procedure is finalized, i.e. that K = ∞. On the purely theoretical level this internal infinity can easily be avoided (in the same way as when proving the countability of a countable set of countable sets), though to organize practical computations on such a basis would be futile. A “heuristic” method of determining ϵ and K was developed while using the method. The method of selecting ϵ described in [1] required no serious modification. A reliable means of selecting K, based on feedback of the computational data, was only discovered later; it proved to be extremely successful, notably because the major part of the algorithm operating time is spent on the B procedure and a poor choice of K involves a lot of wasted time. In addition, the method was used to solve specific problems of linear programming occurring in the numerical solution of optimal control problems [2, 3]. The particular feature of these problems, namely, that they originate “continuously”, demanded some specialization of the method. The type of specialization and a “qualitative” proof of it will be found in Section 2 below. Section 1 concerns the iterative solution of problems of linear programming with a large number of dimensions. The main expressions underlying the process are the same as in [1], but they are worth repeating here because they make the paper easier to read and take up little space.

This paper reports on a retrospective test of a linear programming (LP) technique for planning of medical procedures in preventive health examination (PHE) programs. The technique itself has been published earlier; hence only a brief review is presented here. The test was, in essence, a comparison between a PHE program run along conventional lines (all procedures to all patients) and a PHE program guided by the LP technique. The results showed that the LP technique has several advantages over the conventional mode of operation.

The object of this paper is to outline a framework for model building with space for multiple goals. The planning procedure is briefly described and a distinction is drawn between analytical and decision models. For convenience, exposition is limited to input-output and linear programming (although these examples should not be understood as binding) and a possible approach to comprehensive model building is advanced when these two techniques are properly fused. The importance of explicit statement of goals is stressed in the attempt to rationalize the decision making process.

This paper first describes a theory and algorithms for asymptotic integer programs. Next, a class of polyhedra is introduced. The vertices of these polyhedra provide solutions to the asymptotic integer programming problem; their faces are cutting planes for the general integer programming problem and, to some extent, the polyhedra coincide with the convex hull of the integer points satisfying a linear programming problem. These polyhedra are next shown to be cross sections of more symmetric higher dimensional polyhedra whose properties are then studied. Some algorithms for integer programming, based on a knowledge of the polyhedra, are outlined.

The paper is concerned with optimal control problems for a complex dynamic plant consisting of a number of subsystems having the same final objective and the same constraint on control actions. Complex nonlinear relationships betweem plant coordinates and controls are assumed. A linear programming computing algorithm for approximate solution of such problems is proposed.

This paper develops new duality relations in linear programming which give new economic interpretations to the dual problem. These duality relations are part of a new closed-form solution to an economically interesting class of linear programming problems. A generalization of these results to geometric programming is announced, and potential extensions to integer programming as well as potential applications to stochastic programming are indicated.

Linear programming is a modern mathematical technique useful for solving problems that involve the allocation of scarce resources among alternative ends. This paper discusses the technique and, with the use of a simplified example, illustrates a potential application in dental practice.