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The article presents comments on the management science paper "On Sequencing n Jobs on One Machine to Minimize the Number of Late Jobs," by William L. Maxwell. The paper focused on an integer programming formulation of a one-machine job shop problem. The author contends that Maxwell made in error in proving the validity of an optimal algorithm by applying cutting plane constraints to the problem. The author explains that the problem cannot be solved through traditional cutting plane procedures. The author provides a mathematical model in an attempt to correct the proof.

This paper studies the linear programming problem in which all coefficients (even those of the stipulations matrix) are rational functions of a single parameter t called ‘time,’ and provides an algorithm that can serve problems of the following two types: (1) Steady-state behavior [the algorithm can be used to determine the functional form x(t) of the optimal solution as a function of t, this form being valid for all ‘sufficiently large’ values of t], and (2) sensitivity analysis [if a value t0 of ‘time’ is given, the algorithm can be used to determine the two possible functional forms of the optimal solution for all values of t ‘sufficiently dose’ to t0 (the first functional form valid for t«t0, the second for t»t0)]. In addition, the paper gives certain qualitative information regarding steady-state behavior, including the following result: If for some one of the properties of consistency, boundedness, or bounded constraint set, there exists a sequence tn↗+∞ such that the linear program at tn has this property for all n, then the program has this property for all ‘sufficiently large’ values of t. [ABSTRACT FROM AUTHOR]

The optimum coupon schedule for serial bonds issued by municipalities has been solved as a knapsack problem, and is widely implemented in bank and nonbank underwriting firms. A large subset of issues carries the additional requirement that limits the number of distinct coupons which the underwriters may assign to the issue. The paper formulates this problem as a dynamic programming model and discusses the computational aspects relating to this formulation by comparing it with a direct 0/1 integer programming model. Some computational experience is also provided. [ABSTRACT FROM AUTHOR]

The bond portfolio problem is viewed as a multistage decision problem in which buy, sell, and hold decisions are made at successive (discrete) points in time. Normative models of this decision problem tend to become very large, particularly when its dynamic structure and the uncertainty of future interest rates and cash flows are incorporated in the model. In this paper we present a multiple period bond portfolio model and suggest a new approach for efficiently solving problems which are large enough to make use of as much information as portfolio managers can reasonably provide. The procedure utilizes the decomposition algorithm of mathematical programming and an efficient technique developed for solving subproblems of the overall portfolio model. The key to the procedure is the definition of subproblems which are easily solved via a simple recursive relationship. [ABSTRACT FROM AUTHOR]

This paper will present a model of equipment replacement in a stochastic environment utilizing a number of the component characteristics of an operations research orientation while being designed to emphasize the role of a stochastic representation of technological change and/or maintenance costs and the implications these have for the decision about replacement of equipment. The "replacement" referred to here is replacement of a machine in its entirety rather than the usual periodic-review-replacement-of-individual-pieces analysis to which the term "replacement" frequently refers in operations research literature. The paper outlines the approaches to equipment replacement in the literature of economics and the well-known Terborgh model and then introduces a stochastic model for studying certain aspects of equipment replacement together with the form of its analytic solution. A comparison of the cost implications of the Terborgh and stochastic model is presented for a specific example utilizing motor vehicle data. In addition, a general analytic comparison of the two replacement rules is provided. [ABSTRACT FROM AUTHOR]

This paper is concerned with characterizing decision rules for the sequential E-model of chance-constrained programming. A key feature of our characterization will be a detailed discussion of various interpretations of the probability operator in the chance constraints. Specifically we define two new classes of decision rules by exhibiting those sets of constraints which locally support the corresponding probability requirements. The question of how the probabilistic constraints for future periods are affected by previous decisions and realizations of the random variables is considered in detail. Since we are primarily concerned with the feasibility of decision rules, we deal mainly with the constraints of the model. The procedure for selecting the optimum rule from among a particular class of feasible rules depends on the objective function and is briefly discussed in the final section along with some implications concerning the form of the optimum rule. The application of our proposed rules to a two-period example previously appearing in the literature concludes the paper. [ABSTRACT FROM AUTHOR]

The paper focuses on multi-stage production systems. If such a system is of the more general type than a production line and many products have to be manufactured according to different technological sequence restrictions then two problems arise: (1) scheduling and (2) sequencing. The formulation of that problem requires the simultaneous consideration of the scheduling and sequencing aspect. The paper integrates these two functions into one single model using zero-one programming. [ABSTRACT FROM AUTHOR]

This paper establishes sufficient conditions for decision rules to be optimal for two n-period P-models of chance-constrained programming. The models considered are the triangular model with total probability constraints and the block triangular model with conditional probability constraints. It is shown that, for both these models, a sufficient condition for optimality is that the decision rule be the optimal piecewise linear rule. Proofs of theorems are based on results contained in earlier papers by the authors, on n-period E-models of chance-constrained programming. [ABSTRACT FROM AUTHOR]

An inter-temporal quadratic programming model for selecting portfolios of risky assets is formulated. The model's application to capital budgeting is discussed in considerable detail. This is followed by briefer discussions of its application in other areas. The paper develops a new and more efficient way of using simulation to calculate the variance-covariance elements required as input to Markowitz-type models; this new procedure makes no further demands on the decision maker than do other simulation models that have been suggested. The paper also shows how such models can be made inter-temporal. Finally, the paper analyses the potentials for extending such models to include other theoretical considerations that have been proposed in the literature. [ABSTRACT FROM AUTHOR]

This paper gives a detailed description of an algorithm for the solution of a specialized Linear Programming model, to be called the Machine Loading model. It is a generalization of the Transportation Problem, in that weighting factors are applied to the individual elements which form the row and column sums. For the Machine Loading model, the simplex method reduces to a specialized algorithm which generalizes the stepping stone method of the Transportation Problem [2], [3]. With the resulting generalized stepping stone method, it becomes practicable to solve many large-scale problems for which the direct application of the simplex method would be impracticable. The present paper is restricted to a discussion of the following topics: (I) the general characteristics of the model and its topological features; (ii) a detailed solution algorithm, including a consideration of degenerate cases and the use of a computer; (iii) a more restrictive capacitated model and the corresponding modifications to the solution algorithm; and (iv) the complete illustrative solution of a numerical example. The purpose is to aid interested readers in gaining familiarity with the algorithm and facility in the solution of numerical problems. No derivations or proofs will be included. [ABSTRACT FROM AUTHOR]

This paper is concerned with formulating, exploring and interpreting the uses and constructs which may be derived from a mathematical model of programming type which expresses more realistically than past efforts the actual conditions of current operations. It is an attempt to provide a means of attaining thereby an objective understanding of the implications of actual Federal Reserve liquidity policy on the actions and opportunities of banking institutions. The model presented corresponds to the problem of determining an optimal portfolio for an individual bank over several time periods in accordance with requirements laid down by bank examiners which are interpreted as defining limits within which the level of risk associated with the return on the portfolio is an acceptable one. The problem is transformed into an equivalent one offering advantages of analysis and of computation. Some methods of effective computation are explored. [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 presents a new approach to determining the optimal facility locations for uncapacitated location problems in two stages. First, it is shown that a subset of all solutions to the uncapacitated public facility location problem can be obtained by considering a closely related private location problem. The exact nature of the problem is established using the Generalised Lagrange Multiplier results given by Everett. In the second section of the paper a particularly efficient parametric decomposition algorithm, which is based on the results of the first section, is presented. Computational results are given which summarize the current level of experience with the algorithm. [ABSTRACT FROM AUTHOR]

This paper develops regularity conditions for a class of convex programming problems (convex objective functions and linear constraints). The objective functions considered are lower semicontinuous and have bounded level sets. The constraint set may be unbounded. Results pertaining to the solvability, stability and dualizability of such programs are obtained. The results are then applied to a class of nonlinear stochastic programs with recourse. [ABSTRACT FROM AUTHOR]

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The paper describes a method of representing and analysing the multi-stage, dynamic nature of many types of research and development projects. The method makes use of a number of well-known techniques, decision trees, mathematical programming and simulation in combination. Each project which could form part of a laboratory portfolio is represented in a modified decision tree format called a "project tree", and the project trees are then considered in one analysis based on heuristics. The approach is demonstrated using a small illustrative example, and an industrial case study is described, giving attention to computational problems of size. [ABSTRACT FROM AUTHOR]

This survey examines eight mathematical programming models of pattern classification. The paper determines the range of applicability and computational merits of each model. The flexibility and large sample properties of the models are also discussed. We find that six of the models produce decision rules that maximize a function we call the quality of a decision rule. The remaining two models minimize a weighted sum of errors. [ABSTRACT FROM AUTHOR]

The rapid rate of technological development and the growing complexity of society in recent years have brought renewed awareness of the importance of higher education. The rapid expansion of higher education, both in size and quality, requires systematic approaches and dynamic planning for efficient resource allocation on the part of the university administrators. This paper presents a goal programming model for an optimum allocation of resources in an institution of higher learning. [ABSTRACT FROM AUTHOR]

The author responds to the paper "The Loading Problem," by Eilon and Christofides. He focuses on suggesting a solution to the problem. The recommended solution involves a heuristic algorithm that allegedly requires fewer computations than the one proposed by Eilon and Christofides. It is suggested that the simpler algorithm offered by the author reaches mathematical optimization as often as the aforementioned algorithm. It is further proposed that the algorithm presented by Eilon and Christofides cannot be used to solve multi-dimensional programming problems. The author goes on to discuss a numerical analysis that can be used for the allocation of industrial resources. Related mathematical models are discussed in detail. A table comparing the results of the two algorithms is also presented.

In this paper we present an extension of the simplex procedure to deal with separable programming problems where the objective is composed of functions which are either convex or concave and where the feasible solution set is convex. [ABSTRACT FROM AUTHOR]

A framework of concepts is developed which helps to unify a substantial portion of the literature on large-scale mathematical programming. These concepts fall into two categories. The first category consists of problem manipulations that can be used to derive what are often referred to as "master" problems; the principal manipulations discussed are Projection, Inner Linearization, and Outer Linearization. The second category consists of solution strategies that can be used to solve the master problems, often with the result that "subproblems" arise which can then be solved by specialized algorithms. The Piecewise, Restriction, and Relaxation strategies are the principal ones discussed. Numerous algorithms found in the literature are classified according to the manipulation/strategy pattern they can be viewed as using, and the usefulness of the framework is demonstrated by using it (see Part II of this paper) to rederive a representative selection of algorithms. The material presented is listed in the following order: The first section is introductory in nature, and discusses types of large-scale problems, the scope of discussion and the literature, and the notation used. The second section, entitled "Problem Manipulation: Source of 'Master' Problems" covers the subjects of projection, inner linearization and outer linearization. The third section, "Solution Strategies: Source of 'Subproblems'," discusses piecewise strategy, restriction and relaxation. The fourth section is entitled "Synthesizing Known Algorithms from Manipulations and Strategies," and is followed by a concluding section and an extensive bibliography. [ABSTRACT FROM AUTHOR]

A series of probabilistic models are formulated for the credit granting policy of a firm. First, a single-period analysis is presented for the two-action problem, "give credit or do not give credit." Then a multi-period analysis for the same problem is presented, showing that the single-period analysis ignores important future benefits. The multi-period analysis contains a Bayesian approach to revisions of the probability of collection as collection experience is gained. The multi-period analysis is modified by adding discounting to reflect the time value of money and a probability that the customer will cease purchasing. Next, a dynamic programming formulation of the multi-media problem is presented, and this formulation is then adapted to include a decision on how much credit to offer. The paper closes with a series of remarks concerning the practical application of the analysis. A series of probabilistic models are formulated for the credit granting policy of a firm. First, a single-period analysis is presented for the two-action problem, "give credit or do not give credit." Then a multi-period analysis for the same problem is presented, showing that the single-period analysis ignores important future benefits. The multi-period analysis contains a Bayesian approach to revisions of the probability of collection as collection experience is gained. The multi-period analysis is modified by adding discounting to reflect the time value of money and a probability that the customer will cease purchasing. Next, a dynamic programming formulation of the multi-media problem is presented, and this formulation is then adapted to include a decision on how much credit to offer. The paper closes with a series of remarks concerning the practical application of the analysis. [ABSTRACT FROM AUTHOR]

This paper presents a grouping procedure which yields improved solutions to Bomberger's Lot Size Scheduling Problem [1]. The group definition is a generalization of the group definition proposed by Stankard and Gupta [3]. Moreover, the generalization leads to a pseudo dynamic programming procedure which is readily programmable. Computational experience has shown the procedure to be extremely efficient. [ABSTRACT FROM AUTHOR]

In this paper the Fiacco-McCormick SUMT technique is embedded in a class of inside-out algorithms. Convergence is demonstrated for the nonlinear programming problem under fairly general conditions and the algorithms are interpreted in a geometric structure. [ABSTRACT FROM AUTHOR]

Two dynamic economic lot size production systems are analyzed in this paper, the first being a single product model with backlogging and the second a multi-echelon model. In each model the objective is to find a production schedule that minimizes the total production and inventory costs. A key conceptual difficulty is that the mathematically perplexing problem of minimizing a concave function is being considered, it is shown that both models are naturally represented via single source networks. The network formulations reveal the underlying structure of the models, and facilitate development of efficient dynamic programming algorithms for calculating the optimal production schedules. [ABSTRACT FROM AUTHOR]

The article focuses attention on the problems encountered in formulating and using process mathematical models in an industrial environment. A model of an actual lime factory is used for descriptive purposes. The linear programming technique is employed to illustrate the simulation of the plant. The paper does not try to explain the mathematics of linear programming or any other technique used to simulate the activities of a plant. The aim of the writers is rather to outline the steps taken in formulating a plant model and what can be done with it. This paper is mainly concerned, therefore, with the managerial, engineering and cost accounting problems encountered. The linear programming technique is applied here mainly because of its wide range of applications. Much of what is described may be of value to persons using other methods. [ABSTRACT FROM AUTHOR]

This paper demonstrates the equivalence of several published algorithms for solving the so-called linear fractional programming problem. [ABSTRACT FROM AUTHOR]

A finite horizon, periodic review, dynamic inventory model for a single item with a stockout constraint is considered in this paper, and the optimal ordering policy is obtained. The discrete case is dealt with, and linear purchase and holding costs are assumed together with a fixed order cost, all costs being discounted back to the beginning of the first period. A fixed delivery lead time is assumed, and backlogging of unsatisfied demand is required. The stockout constraint is that the ratio of the expected number of stockouts to the expected demand over the finite horizon not exceed a preassigned fraction. The optimal ordering policy is obtained by the usual dynamic programming technique, using a Lagrange multiplier to handle the stockout constraint. [ABSTRACT FROM AUTHOR]

In linear programming under uncertainty the two-stage problem is handled by assuming that one chooses a first set of constrained decision variables; this is followed by observations of certain random variables after which another set of decisions must be made to adjust for any constraint violations. The objective is to optimize an expected value functional defined relative to the indicated choices. This paper shows how such problems may always be replaced with either constrained generalized medians or hypermedians in which all random elements appear only in the functional. The resulting problem is called a deterministic equivalent for the original problem since (a) the originally defined objective replaces all random variables by corresponding expected values and (b) the remaining constraints do not contain any random terms. Significant classes of cases are singled out and special attention is devoted to the structure of the constraint matrices for these purposes. Numerical examples are supplied and related to the previous literature. Other properties of these models are also examined and related to types of problems which are often of interest. For instance the hypermedian and generalized median formulations involve minimizations over absolute value terms in the functional. These, in turn, are developed for their possible pertinence in problems where minimizations are to be over the maximum of a set of functions under inequality constraints. Utilizing Moore-Penrose (generalized) inverses, other characterizations are also secured in which all relevant weights and coefficients are stated explicitly in terms of original data. [ABSTRACT FROM AUTHOR]

This paper shows in a hypothetical and illustrative manner, how a linear programming model of moderate size can be used in selecting projects for a national development program, without requiring the construction of a model that covers the whole of the economy in detail. [ABSTRACT FROM AUTHOR]

This paper describes an algorithm for the solution of mathematical programming problems having a linear objective function and non-linear constraints. The algorithm is basically an adaptation of the simplex method to the case of non-linear constraints. Certain complications are, however, introduced through non-linearity in the constraints, and it is shown how these can be overcome. [ABSTRACT FROM AUTHOR]

As interest grows rapidly in industry on the potentialities of mathematical programming techniques, it appears worthwhile to have a paper devoted to some of the more promoting developments which may speed up the transition from interest to use. Three topics have been selected (in three sections that follow) which have recently come into prominence: uncertainty, combinatorial problems, and large scale systems. The reader will find in the course of their dissensions that a survey--though perhaps not a systematic survey--has been made of current techniques in the linear programming field. [ABSTRACT FROM AUTHOR]

This paper considers a stationary discrete dynamic programming model that is a generalization of the finite state and finite action Markov programming problem. We specify conditions under which an optimal stationary linear decision rule exists and show how this optimal policy can be calculated using linear programming, policy iteration, or value iteration. In addition we allow the parameters of the problem to be random variables and indicate when the expected values or these random variables are certainty equivalents. [ABSTRACT FROM AUTHOR]

A multi-stage linear program is defined with linking variables that connect consecutive stages. Optimality conditions for the composite problem are partitioned into local and linking conditions. When the Dantzig-Wolfe decomposition scheme is applied with the first stage as the master, the subproblem is also a MLP with one less stage. The same decomposition is then applied to the subproblem, giving rise to a nested decomposition scheme, in which each stage acts as a master for the following stage and a subproblem for the preceding. Optimizing a single stage problem results in satisfying the "local" optimality conditions. A very general rule is given for selecting the next subproblem to optimize, and finite convergence to a solution satisfying all linking conditions is demonstrated. Procedures for extracting the optimal primal solution at the end of the main algorithm and for initialization are given. Particular rules for selecting the next subproblem and for generating additional proposal vectors are discussed. Finally, it is shown how a variety of problems may be restructured as multi-stage linear programs to which this algorithm may be applied, and some computational experience is reported. [ABSTRACT FROM AUTHOR]

This paper is written with three objectives in mind. First, to point out that the problem of grouping, where a larger number of elements n are combined into m mutually exclusive groups (m ~n) should be recognized as a problem in Integer Programming, and that such recognition can help us in avoiding complete enumeration of stages in grouping from n to n - 1 ... to m, and of alternative possibilities in each stage. Second, to formulate mathematically some simple versions of the relevant Integer Programming Problem, so that the available computer codes can solve it. When the grouping attempts to minimize the within groups sums of squares, the so-called string property is proved to be necessary for the minimum (Lemma 1). It is shown that the string property can be exploited to write the non-linear within group sums of squares as a linear function (Lemma 2). An attempt is made to generalize the string property for the higher dimensional case. Third, to give some numerical examples to clarify the mathematical models and to illustrate the advantages of Integer Programming formulation by comparing with the numerical examples appearing in the literature [8], [13 ]. Zero-one programming where the integer variables can take values of zero or one only appears to be most efficient from the limited experience of the author. The paper is divided into three sections which closely correspond to the three objectives noted above. [ABSTRACT FROM AUTHOR]

This paper considers the problem of determining the policy to be followed by each member of a team so as to optimize some over-all criterion function characterizing the objective of the team. The reasoning of dynamic programming is employed to formulate the optimization problem, and the discussion is couched in the terms of a classical network problem. The paper first formulates a simple bipartite network optimization and then presents a number of generalizations. Finally, it gives two slightly different computational solutions to an example bipartite optimization problem. [ABSTRACT FROM AUTHOR]

The growth and increasing complexity of municipal governments have been phenomenal in recent years. Thus, economic planning of the municipality has become one of the most difficult administrative functions of local government officials. Yet economic systems analysis of cities has been generally neglected by economists and management scientists. This paper presents a methodology for the effective long-range economic planning of a municipal government. More specifically, it applies the goal programming model to design optimum aggregative model for municipal economic planning. [ABSTRACT FROM AUTHOR]

Variants of Markovian decision models are many. Of those concerned with the estimation of transition probabilities, few deal with the case of limited information in which only data on market shares are available. Where such investigations have been undertaken, the assumption of constant transition probabilities has been insurmountable. This paper offers a simple heuristic technique to estimate nonstationary transition probabilities, especially in the case where fluctuations in market shares are felt to reflect the effects of secondary considerations external to the transition probabilities as well as effects of the probabilities themselves. Dynamic and quadratic programming are the suggested tools of analysis. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with imposing constraints directly on the admissible majority decisions so as to insure transitivity without restricting individual preference orderings. We demonstrate that this corresponds to requiring that majority decisions be confined to the extreme points of a convex polyhedron. Thus, transitive majority decisions can be characterized as basic solutions of a set of linear inequalities. Through the use of a majority decision function (which is not restricted to be linear) it is shown that constrained majority rule is equivalent to an integer programming problem. Some special forms of majority decision functions are studied ineluding the generalized lp norm and an indicator function. Implications of an integer programming solution, including alternate optima and post optimality analysis, are also discussed. [ABSTRACT FROM AUTHOR]

It has been established that integer programs with integral, recursive constraint matrices have integral extreme points for integral, nonnegative requirement vectors. In this paper, it is first shown that the assignment problem can be transformed to such a program, but with upper bounds on some variables. This equivalent program leads to additional results for solving the assignment problem. Algorithms are then delineated to transform matching, covering and travelling salesman problems to equivalent recursive programs of the aforementioned type, but with bounds on some variables. These results are extended to programs with special structure in their constraint matrices. [ABSTRACT FROM AUTHOR]

In this paper we discuss the class of stochastic programs with recourse in which the only randomness present is in the recourse costs. Two economic interpretations are given. We present results which enable one to check if the convex deterministic equivalent program possesses all the "nice" properties of convex programs -- feasibility, boundedness, solvability and dualizability. The simple recourse problem is then discussed. The specific form of the convex deterministic equivalent for the case of simple recourse is exhibited. [ABSTRACT FROM AUTHOR]

The multiparametric linear programming (MLP) problem for the right-hand sides (RHS) is to maximize z = cT x subject to Ax = b (λ), x ≧ 0, where b(λ) can be expressed in the form b(λ) = b* + Fλ, where F is a matrix of constant coefficients, and λ is a vector-parameter. The multiparametric linear programming (MLP) problem for the prices or objective function coefficients (OFC) is to maximize z = cT (v)x subject to Ax = b, where x ≧ 0, where c(v) can be expressed in the form c(v) = c* + Hv, and where H is a matrix of constant coefficients, and v a vector-parameter. Let Bi be an optimal basis to the MLP-RHS problem and Ri be a region assigned to Bi such that for all λ Ri the basis, Bi is optimal. Let K denote a region such that K = ui Ri , provided that the Ri for various i do not overlap. The purpose of this paper is to present an effective method for finding all regions Ri that cover K and do not overlap. This method uses an algorithm that finds all nodes of a finite connected graph. An analogous method is presented for the MLP-OFC problem. [ABSTRACT FROM AUTHOR]

Recent efforts in the area of modelling activities in the urban environment have been concentrated on the use of simulation techniques. In this paper the problem of population location in a metropolitan area is treated in an analytic model employing a goal programming formulation. A target population is broken into groups which then compete for a share of limited capacity in a number of urban sectors. The possible usefulness of this approach to urban planning, real estate development, and transportation systems design is illustrated through the theoretical development and the presentation of a numerical model. [ABSTRACT FROM AUTHOR]

Growth period models, previously treated in the literature, have assumed that the pattern of value increase of the growth asset is deterministic. In this paper, this assumption is relaxed by considering models in which the increase in value of an asset in a period is a random variable whose distribution is a function of either the value or the age of the asset at the start of the period. The expected increase in value is a decreasing function of the value or age of the asset so that the value of additional maturation time decreases as the asset ages. Dynamic programming is used to compute optimal policies as to when stochastic growth assets should be harvested. The steady state value function is shown to be directly analogous to that obtained when deterministic growth is assumed. Procedures for quickly computing both steady state policies and value functions are developed. [ABSTRACT FROM AUTHOR]

The objective of this paper is the construction of a branch-and-bound algorithm for computing minimum cost flows in capacitated networks when the cost of passing a flow over an arc is concave. The method is based on the equivalence of the general network flow problem to a network flow problem in a bipartite network of a special form. An optimal flow is found by implicit enumeration of the set of extremal flows in that network. [ABSTRACT FROM AUTHOR]

In this paper a process analysis model of production in the U. S. Steel Industry is developed and tested. A technology matrix, which represents recent industry input-output structure, is estimated using engineering and metallurgical information. This matrix together with detailed cost, sales and resource capacity data is then used in a linear programming model of short-run resource allocation for the industry as a whole. Linear programming solutions are obtained and compared with available industry statistics for each year in the period 1955-1968. Theil's U statistic is computed to indicate how well the model performs. We conclude that it performs "fairly well" and that it should prove useful in various economic applications. [ABSTRACT FROM AUTHOR]

From a narrow point of view, linear programming is a method for calculating optimal operations plans for various corporate activities. From a broader point of view, however, the theory of linear programming provides a basis for the study of the economics of corporate activity. See, e.g., [1] For example, by providing existence theorems it is possible in many instances to say whether a proposed operation is possible at all; by providing characterization theorems it is possible to identify which variables are important and which are not, and what their relationships must be for optimal operation. The case of nonlinear convex programming is similar. Narrowly viewed, it provides a basis for the calculation of optimal operations plans. Broadly viewed, the theory ought to provide an even broader insight into the economics of corporate activity, but until recently no really satisfactory mathematical theory has been available. The work of R. T. Rockafellar [4], [5] now appears to fill this need, and in this paper we begin the development of a more general theory of the economics of corporate activity, from the activity analysis [3] point of view. [ABSTRACT FROM AUTHOR]

The decomposition of a finite partially ordered set of elements as a union of chains was considered by Dilworth [2]. Dantzig and Hoffman [1] formulated this problem as a linear programming problem and obtained Dilworth's theorem from duality theory. For some practical applications and for a method to obtain a minimal decomposition see Ford & Fulkerson [3]. In this paper we generalise this problem to the case when the set is not necessarily partially ordered and obtain a method of finding a minimal chain decomposition of the set. [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]

This paper presents an algorithm for the solution of the knapsack problem. The method involves searching the nodes of a tree along a single branch at a time. The algorithm eliminates the computational drawbacks inherent in the usual branch and bound schemes. [ABSTRACT FROM AUTHOR]