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This article presents abstracts of several papers about economics published in the July 1965 of the journal Zeitschrift Für Die Gesamte Staatswissenschaft. In the article The Optimal Location of Industry: An Interregional Linear Program for the West German Paper Industry, Heinz König and Rainer Thoss gave a general formulation of an interregional optimization model and discussed the advantages and limitations of this method of regional analysis. While in the article Development Theories of Balanced and Unbalanced Growth: A Confrontation, Lutz Hoffman reviewed the discussion on balanced growth and unbalanced growth.

This paper gives a framework for unifying and slightly generalizing certain properties of extreme-point solutions exploited by an approach of LEMKE, SALKIN, and SPIELBERG for solving set-covering problems. [ABSTRACT FROM AUTHOR]

A letter to the editor is presented in response to the article "Assembly-Line Balancing by Linear Programming," by E. H. Bowman in a 1960 issue of "Operations Research 8."

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.

The article presents a zero-one linear program and a dynamic programming algorithm for finding the shortest route containing exactly q arcs from node 1 to node n in a network (N, A) with distances c(i, j). This note shows that the linear programming formulation and his extension based on it are defective, and that the dynamic programming algorithm can lead to suboptimal solutions, but a minor change in the dynamic programming formulation relieves the difficulty. A special feature of this linear program is that there can never be a loop in the basis, because the vectors corresponding to the variables of a loop are linearly dependent. The last comment is related to the dynamic programming algorithm. One is allowed to pass through a node more than once in the shortest route . The proposed method for solving this program consists of converting it into a shortest route problem containing exactly q arcs. But for some arbitrary reason, the dynamic program is formulated in such a way that one can never have starting node 1 more than once in the final solution.

In this paper we present an outline of LONGER, a financial planning model based on mixed integer linear programming. The model is based on recent advances in capital market theory, but at the same time it recognizes certain additional considerations that are manifestly important to the financial manager. Thus, we believe that the model has good potential for practical application, although in this paper we emphasize LONGER's relationship to finance theory. We will skim over a number of issues that are critical in a practical application; a later paper will discuss these issues and present LONGER at a practical level of detail. Still another paper has worked out implications for decentralized capital budgeting rules. [ABSTRACT FROM AUTHOR]

With fluctuating sales, a manufacturer must have fluctuating production, or fluctuating inventory, or both Penalties are associated with either type of fluctuation Several papers place this problem into a conventional linear-programming framework This paper suggests that the same problem may be placed into a transportation-method framework and, further, that many transportation problems may be extended to include multiple time periods where this is meaningful A generalized scheduling problem is placed here into the standard form of the transportation table [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to delineate an algorithm for fractional programming with nonlinear as well as linear terms in the numerator and denominator. The algorithm presented is based on a theorem by Jagannathan [7] concerning the relationship between fractional and parametric programming. This theorem is restated and proved in a somewhat simpler way. Finally, it is shown how the given algorithm can be related to the method of Isbell and Marlow [6] for linear fractional programming and to the quadratic parametric approach by Ritter [10]. The Appendix contains a numerical example. [ABSTRACT FROM AUTHOR]

Commentary is presented for an article published in the July 1960 issue of "Management Science," written by L. V. Kantorovich with an introductory note by T. C. Koopmans. According to the author, in his introduction Koopmans addresses in particular a linear programming problem and a matrix game presented by Kantorovich. Also presented is an interpretation of the article's treatment of decision variables and mathematical optimization. The author notes that the article presents algorithms that have been improved from their contemporary presentations.

This paper is the latest chapter in the controversy begun by William H. Bautool and Richard E. Quandt in their criticism of Weingartner's work on capital budgeting under capital rationing. A thorough review and partial resolution of the matter has been offered in recent articles by Willard T. Carleton and Edwin J. Elton. My purpose here is to try to complete the job. [ABSTRACT FROM AUTHOR]

One of the most impressive achievements in capital budgeting theory during recent years has been by H. Martin Weingartner, who successfully employed the technique of linear (and integer linear) programming to solve the "capital budgeting under capital rationing" problem. A subsequent debate between Weingartner and William J. Baumol and Richard E. Quandt, however, has suggested the possibility that programming approaches create more questions than solutions, it is the thesis of this paper that some of the major issues between Weingartner and Baumol-Quandt can be resolved easily through appeal to the larger corporate financial model of which project selection is but one part. In particular, it is possible to suggest what is the appropriate objective function to be employed in selecting a set of projects; what is the best intellectual construction and operating significance of the term "capital rationing"; and what is the relationship under capital rationing of the discounting rate to the size of the capital budget. [ABSTRACT FROM AUTHOR]

This article presents comments on a paper by M.L. Wolfson related to selecting the best lengths to stock. The problem considered in the paper by Wolfson is a classic specimen of the type ideally structured for direct and practical solution by dynamic programming. The author's algorithm, given in the paper, is in the spirit of this method, but its understanding could be greatly simplified if put into the recursive formalism of dynamic programming.

This paper describes the application of a linear programming model to the process of resource allocation at Illinois State University. The model developed is a general model designed to assist academic administrators in allocating the scarce resources at their disposal. The model, and the assumptions upon which it is based, may be altered to suit the particular decision criteria of any particular academic institution. The linear programming model developed here maximizes an objective function which reflects the total net social profit realized by the university on its outputs to society. This objective function was maximized subject to a myriad of constraints which reflected the limitations of budget, space, and other scarce inputs. Further, the constraints reflect the joint interdependency of many parts of the university and the particular educational delivery system employed by a given department. The outputs of the model include the optimal mixture of academic outputs by the university, the optimal distribution and usage of inputs inside the university, and the shadow price of each input. While the linear programming model developed here is general and may be applied to any university, it was applied to Illinois State University in this paper. In the case of Illinois State University, the model recommended a substantial reduction in emphasis upon teacher education. [ABSTRACT FROM AUTHOR]

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]

This paper establishes some useful properties of the equality-constrained set-covering problem P and the associated linear program P'. First, the Dantzig property of transportation matrices is shown to hold for a more general class of matrices arising in connection with adjacent integer solutions to P'. Next, we show that, for every feasible integer basis to P', there are at least as many adjacent feasible integer bases as there are nonbasic columns. Finally, given any two basic feasible integer solutions x¹ and x² to P', x² can be obtained from x¹ by a sequence of p pivots (where p is the number of indices j∈N for which x¹[subj] is nonbasic and x¹[subj]=1), such that each solution in the associated sequence is feasible and integer. Some of our results have been conjectured earlier by ANDREW, HOFFMAN, AND KRABEK in a paper presented to ORSA in 1968. [ABSTRACT FROM AUTHOR]

This paper considers the mathematical properties of chance constrained programming problems where the restriction is on the joint probability of a multivariate random event. One model that is considered arises when the right-hand side constants of the linear constraints are random. Another model treated here occurs when the coefficients of the linear programming variables are described by a multinormal distribution. It is shown that under certain restrictions both situations can be viewed as a deterministic nonlinear programming problem. Since most computational methods for solving nonlinear programming models require the constraints be concave, this paper explores whether the resultant problem meets the concavity assumption. For many probability laws of practical importance, the constraint in the first type of model is shown to violate concavity. However, a simple logarithmic transformation does produce a concave restriction for an important class of problems. The paper also surveys the `generalized linear programming' method for solving such problems when the logarithmic transformation is justified. For the second type model, the constraint is demonstrated to be nonconcave. [ABSTRACT FROM AUTHOR]

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

Presents abstracts of invited and contributed papers published in the fall 1961 issue of the journal 'Operations Research.' 'Experiments in Linear Programming,' by Philip Wolfe; 'Programs With Linear and Square Root of Quadratic Forms in the Objective,' by E. Eisenberg; 'Some Results and Problems in Linear Programming Under Uncertainty,' by Albert Madansky.

The article presents information on the program of the Eight Annual Meeting of the Operations Research Society of America held in New York City from May 18 to 21, 1960. As part of the meeting, the symposium titled Optimization on Chemical Reactors was held at New York University. John Happel served as chairman of the symposium. There were many papers presented during the meeting. A. V. Butterworth and P.C. Hayes presented their paper which discusses the use of mathematical predictors in a distribution system. Other topics discussed at the meeting include linear programming, strategic planning, and production planning.

Although the earliest discussions of linear programming date back to the 1930's, to the work of German and Austrian mathematicians and economists, the mathematics has been extensively developed only in the past six or eight years. Because of the relative newness of the subject, the literature is widely scattered and it is rather difficult for one just starting to learn about linear programming to know where to begin. Because of human limitations, the bibliography is necessarily incomplete. All papers and books that could be acquired that seemed to deal with the subject were perused and, if applicable, included. No extensive effort was made, however, to include papers dealing more or less exclusively with closely related topics, for example, the problem of determination of value and optimal strategies in a zero-sum, two-person game, the pure inequalities problem with no optimization, the subjects of convex functions and convex polyhedra, unless they also relate these topics directly to linear programming.

This paper introduces an efficient branch and bound algorithm for a special class of mixed integer programming problems called the warehouse location problem. A set of branching decision rules is proposed for selecting warehouses to be constrained open and closed from any node of the branch and bound tree. These rules are tested for their efficiency in reducing computation times and storage requirements to reach optimal solutions. An improved method of solving the linear programming problems at the nodes which substantially reduces the computations is also introduced in this paper. [ABSTRACT FROM AUTHOR]

The paper proposes the use of linear programming to "solve" a specific freight car allocation situation wherein a choice of freight cars is available for shipper shipments, and the railroad's car allocation objective is to maximize the expression (freight car rental receivables-freight car rental payables). The freight car rental payment structure which is applicable is that of "time-mileage per diem" rates proposed in the United States by the Interstate Commerce Commission. The paper first describes the problem and its background, and then a model and various extensions are formulated. An example illustrates how costly to a railroad the difference between good and bad car allocation may be. Although the proposed linear programming approach may be manually tractable for small volume car allocation situations, the paper concludes that in most car allocation situations, the approach requires a computerized data base with a "real-time" information and communications system which includes order entry, car rental, routing-distance, and train travel time and yard delay time information. [ABSTRACT FROM AUTHOR]

In an earlier paper [5], we generalized and extended Beckmann's results [1] for a production and inventory problem with proportional smoothing costs and demands being random variables. Our previous results concerned the finite horizon nonstationary case. Here we consider the infinite horizon stationary case. Two curves in the plane determine an optimal policy. They are shown to have slopes between minus one and zero, to be differentiable, and to be bounded by two straight lines with a slope Of minus one. These results are used (a) to accelerate each iteration of a successive approximations algorithm and (b) to formulate a linear programming problem from whose solution an optimal policy can be determined. [ABSTRACT FROM AUTHOR]

This paper describes a simulation model of capital budgeting under uncertainty. It analyzes the effects of two types of uncertainty which influence the cash flows of the potential investment projects. Techniques of simulation and stochastic linear programming (using Weingartner's Basic Horizon model of capital budgeting) are employed to compute the expected return (with an associated measure of the risk involved) of different portfolios of projects. The manager can then select the portfolio which is closest to his personal risk-return preference. The model provides a practical guide to management in the entire process of project search, portfolio generation, and portfolio evaluation which characterizes the capital budgeting decision. [ABSTRACT FROM AUTHOR]

The paper presents a method for solving the linear programming problems, which is itself a step towards the generalization of the classical Simlex Method. It replaces two basic variables by two non-basic variables at each iteration. Naturally this will reduce the total number of steps to reach the final solution. The paper also includes the solution of one example to illustrate the applicability of the method. [ABSTRACT FROM AUTHOR]

A new conceptual and analytical vehicle for problems of temporal planning under uncertainty, involving determination of optimal (sequential) stochastic decision rules is defined and illustrated by means of a typical industrial example. The paper presents a method of attack which splits the problem into two non-linear (or linear) programming parts, (i) determining optimal probability distributions, (ii) approximating the optimal distributions as closely as possible by decision rules of prescribed form. [ABSTRACT FROM AUTHOR]

A mathematical program with a maximin objective function is defined as an optimization problem of the following type: Maxz=min[sub i] c[sub i]x[sub i], subject to AX = b, X * (This character cannot be converted in ASCII text) O. Although the c[sub i] can be in the interval (-- ∞, ∞), the paper discusses the more common practical case where all c[sub t] * (This character cannot be converted in ASCII text) 0. It shows that problems of this type arise in a variety of applications where it is required to maximize a production function of the 'fixed proportion' type subject to a set of linear constraints. Although it is well known that the solution to this type of problem can be found by linear programming, this paper shows that, if the existence of a certain condition can be demonstrated, then a simplified method can be used to determine the optimum solution. Many problems of practical interest can be solved by this simplified method; an example involving the readiness of a ship is presented. [ABSTRACT FROM AUTHOR]