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Although it is easy to prove the sufficient conditions for optimality of a linear program, the necessary conditions pose a pedagogical challenge. A widespread practice in deriving the necessary conditions is to invoke Farkas' lemma, but proofs of Farkas' lemma typically involve "nonlinear" topics such as separating hyperplanes between disjoint convex sets, or else more advanced LP-related material such as duality and anti-cycling strategies in the simplex method. An alternative approach taken previously by several authors is to avoid Farkas' lemma through a direct proof of the necessary conditions. In that spirit, this paper presents what we believe to be an "elementary" proof of the necessary conditions that does not rely on Farkas' lemma and is independent of the simplex method, relying only on linear algebra and a perturbation technique published in 1952 by Charnes. No claim is made that the results are new, but we hope that the proofs may be useful for those who teach linear programming.

I want to write about what I know and remember about the activities of Leonid Vital'evich Kantorovich, an outstanding scientist of the 20th century; about his dramatic struggle for recognition of his mathematical economic theories; about the initial stage of the history of linear programming; about beautuful Kantorovich metric, about the creation of a new area of mathematical activity related to economic applications, which is called sometimes operation research, sometimes mathematical economics, sometimes linear and convex programming, or economic cybernetics, etc.; about its place in the modern mathematical landscape; and, finally, about several personal impressions of this distinguished scientist. The notes in no way pretend to exhaust these topics.

We illustrate the utility of the recently developed loop calculus for improving the Belief Propagation (BP) algorithm. If the algorithm that minimizes the Bethe free energy fails we modify the free energy by accounting for a critical loop in a graphical representation of the code. The log-likelihood specific critical loop is found by means of the loop calculus. The general method is tested using an example of the Linear Programming (LP) decoding, that can be viewed as a special limit of the BP decoding. Considering the (155,64,20) code that performs over Additive-White-Gaussian-Noise channel we show that the loop calculus improves the LP decoding and corrects all previously found dangerous configurations of log-likelihoods related to pseudo-codewords with low effective distance, thus reducing the code's error-floor.

We illustrate the utility of the recently developed loop calculus for improving the Belief Propagation (BP) algorithm. If the algorithm that minimizes the Bethe free energy fails we modify the free energy by accounting for a critical loop in a graphical representation of the code. The log-likelihood specific critical loop is found by means of the loop calculus. The general method is tested using an example of the Linear Programming (LP) decoding, that can be viewed as a special limit of the BP decoding. Considering the (155,64,20) code that performs over Additive-White-Gaussian-Noise channel we show that the loop calculus improves the LP decoding and corrects all previously found dangerous configurations of log-likelihoods related to pseudo-codewords with low effective distance, thus reducing the code's error-floor.

This volume was digitized and made accessible online due to deterioration of the original print copy.

Linear programming provides methods for computing the most practical solutions in production, operations, and supply planning and in the control of complex processes. The introduction of linear programming into practice materially cuts costs and time losses. The present book presents in detail the mathematical theory of linear programming and computational methods yielding an exact solution over a finite number of steps. The book is intended for engineers, economists, and applied mathematicians. It may also be used by university students in the mathematics, economics, and engineering economics departments. This book is a translation of LINEINOE PROGRAMMIROVANIE Teoriya i konechnye metody Gosudarstvennoe Izdatel’stvo Fiziko-Matematicheskoi Literatury Moskva 1963

Linear programming provides methods for computing the most practical solutions in production, operations, and supply planning and in the control of complex processes. The introduction of linear programming into practice materially cuts costs and time losses. The present book presents in detail the mathematical theory of linear programming and computational methods yielding an exact solution over a finite number of steps. The book is intended for engineers, economists, and applied mathematicians. It may also be used by university students in the mathematics, economics, and engineering economics departments. This book is a translation of LINEINOE PROGRAMMIROVANIE Teoriya i konechnye metody Gosudarstvennoe Izdatel’stvo Fiziko-Matematicheskoi Literatury Moskva 1963

Three topics: uncertainty, combinatorial problems, and large scale systems are discussed.

The multi-armed bandit problem and one of its most interesting extensions, the restless bandits problem, are frequently encountered in various stochastic control problems. We present a linear programming relaxation for the restless bandits problem with discounted rewards, where only one project can be activated at each period but with additional costs penalizing switching between projects. The relaxation can be efficiently computed and provides a bound on the achievable performance. We describe several heuristic policies; in particular, we show that a policy adapted from the primal dual heuristic of Bertsimas and Nino-Mora [1] for the classical restless bandits problem is in fact equivalent to a one-step look ahead policy; thus, the linear programming relaxation provides a means to compute an approximation of the cost-to-go. Moreover, the approximate cost-to-go is decomposable by project, and this allows the one-step look ahead policy to take the form of an index policy, which can be computed on-line very efficiently. We present numerical experiments, for which we assess the quality of the heuristics using the performance bound.

The multi-armed bandit problem and one of its most interesting extensions, the restless bandits problem, are frequently encountered in various stochastic control problems. We present a linear programming relaxation for the restless bandits problem with discounted rewards, where only one project can be activated at each period but with additional costs penalizing switching between projects. The relaxation can be efficiently computed and provides a bound on the achievable performance. We describe several heuristic policies; in particular, we show that a policy adapted from the primal dual heuristic of Bertsimas and Nino-Mora [1] for the classical restless bandits problem is in fact equivalent to a one-step look ahead policy; thus, the linear programming relaxation provides a means to compute an approximation of the cost-to-go. Moreover, the approximate cost-to-go is decomposable by project, and this allows the one-step look ahead policy to take the form of an index policy, which can be computed on-line very efficiently. We present numerical experiments, for which we assess the quality of the heuristics using the performance bound.

From the bitsavers.org collection, a scanned-in computer-related document. ibm :: 1620 :: general program library :: 10.1.007 Linear Programming I

From the bitsavers.org collection, a scanned-in computer-related document. ibm :: 1620 :: general program library :: 10.1.009 LP20 Linear Programming System

Khachian's polynomial bound for finding a feasible solution to a relaxed linear program is an important theoretical result. Unfortunately, a polynomial bound does not imply a good algorithm because such a bound could be too large for problems of practical interest. For example, using the formulae in the original paper, practical problems with 3000 non-negative variables and 1000 equations (which are solved under one-half hour on IBM 370-168 using the simplex method) would involve over 10 to 15th power iterations and would take 50,000,000 years to solve using Khachian's method.

The United States Armed Forces must be capable of deploying to areas of operations anywhere is the world. Planning for these deployments is the responsibility of the Joint Deployment Agency, MacDill Air Force Base, Tampa, Florida. Deployment plans are large and complex. A straightforward linear programming model of a deployment plan could easily exceed 700 million decision variables. This study outlines the development of a system used to assist planners in determining deployment plan feasibility and in selecting modes of transportation. The system consists of a data input array, an algorithm to eliminate all unusable variables, and a linear programming model. The largest scenario in this study is a 90-day deployment plan with 80 movement requirements, 9 types of lift assets, traveling between 22 ports. This corresponds to a linear programming model with 35 million decision variables. The variable reduction algorithm reduced the number of variables to 11,100, and an optimal solution was found in a total computation time (input, reduction, optimization, output) time of 6.5 minutes.

The United States Armed Forces must be capable of deploying to areas of operations anywhere is the world. Planning for these deployments is the responsibility of the Joint Deployment Agency, MacDill Air Force Base, Tampa, Florida. Deployment plans are large and complex. A straightforward linear programming model of a deployment plan could easily exceed 700 million decision variables. This study outlines the development of a system used to assist planners in determining deployment plan feasibility and in selecting modes of transportation. The system consists of a data input array, an algorithm to eliminate all unusable variables, and a linear programming model. The largest scenario in this study is a 90-day deployment plan with 80 movement requirements, 9 types of lift assets, traveling between 22 ports. This corresponds to a linear programming model with 35 million decision variables. The variable reduction algorithm reduced the number of variables to 11,100, and an optimal solution was found in a total computation time (input, reduction, optimization, output) time of 6.5 minutes.

The United States Armed Forces must be capable of deploying to areas of operations anywhere is the world. Planning for these deployments is the responsibility of the Joint Deployment Agency, MacDill Air Force Base, Tampa, Florida. Deployment plans are large and complex. A straightforward linear programming model of a deployment plan could easily exceed 700 million decision variables. This study outlines the development of a system used to assist planners in determining deployment plan feasibility and in selecting modes of transportation. The system consists of a data input array, an algorithm to eliminate all unusable variables, and a linear programming model. The largest scenario in this study is a 90-day deployment plan with 80 movement requirements, 9 types of lift assets, traveling between 22 ports. This corresponds to a linear programming model with 35 million decision variables. The variable reduction algorithm reduced the number of variables to 11,100, and an optimal solution was found in a total computation time (input, reduction, optimization, output) time of 6.5 minutes.

The paper discusses a particular product form of inverse which is closely related to the Gaussian elimination method of solving a set of simultaneous equations. This 'elimination form of the inverse' is especially valuable when A has a large number of zero coefficients. If A has no zero coefficients, on the other hand, the elimination form of inverse is still generally as convenient as the conventional A (-1).

Dantzig and Van Slyke (AD-610 950) have proved a theorem that gives an upper bound on the number of sets containing at least two basic variables. This fact is exploited to develop an algorithm that is computationally efficient for a large scale system with a special structure. It is possible to show that this bound can be improve. The significance of this improvement lies in the fact that since it is related to the order of the working basis, the computation can be carried out with a basis of order less by one than that considered in the above. The purpose of this note is to describe it and illustrate by an example.

Ralph Gomory has recently aroused interest in a special type of knapsack problem in which the constraint coefficients and constant term are elements of a finite additive group. The significance of this problem lies in the fact that it is closely related to the general integer linear programming problem, resulting by removing the nonnegativity restrictions on those variables in the general problem that lie in an optimal basis for the associated linear program. Gomory has shown how to solve the special knapsack problem by adapting a dynamic programming recursion originally designed for the ordinary knapsack problem, and has identified sufficient conditions under which the solution of the special knapsack problem will satisfy the nonnegativity requirements in the general integer program, thereby yielding an optimal solution to that problem as well. In this paper the author presents an algorithm for solving the special knapsack problem that is capable of accommodating a variety of constraints in addition to the special knapsack constraint. The purpose in doing this is to expand the range of problems for which the optimal solution for the special problem will also provide an optimal solution to the general integer program from which it was derived.

An algorithm has been developed which uses the complementary slackness principle to take completely arbitrary primal and dual solutions to a linear program with doubly-bounded variables into the optimal solutions. The algorithm is not a new method; viewed in the proper context, it can be thought of either as an elaboration of the primal-dual, composite, breakpoint-tracing, or complementary pivot algorithms: as an extension of the out-of-kilter or black-box methods for network flows; or, finally, even as a special way of looking at the original simplex algorithm. Almost all of the simpler procedures, such as Phase I, the dual simplex method, parametric programming, the primal-dual algorithm, etc. can be viewed as special cases of the complex algorithm which use special starting solutions and special heuristics.

From its inception Linear Programming was envisioned as being applied to large detailed dynamic models of economic and industrial systems. Difficulties of obtaining input data, making use of detailed output data, and the cost of computation have in the past limited applications. Three types of approaches have been proposed for efficient computation. These are reviewed in terms of typical matrix structures to which they are applicable. A list of 128 references is appended. (Author)

The Engineer Functional Components System is used by the Army in planning construction of facilities and installations. This operational system employs numerous and extensive tape files containing bills of material for several hundred types of military installations comprised of thousands of items of material. The linear programming models discussed herein use this system in formulating linear constraints on construction capabilities due to resource limitations, and linear criterion functions in terms of the various facilities and/or installations to be constructed. The models provide a capability to rapidly consider alternative construction programs. Single-period and multiperiod models are given, including modifications of the basic models to make them applicable to a wider range of problems.

It is known that the Mizuno-Todd-Ye predictor-corrector primaldual Newton interior-point method generates a duality-gap sequence which converges quadratically to zero, and this is accomplished with an iteration complexity of O (square root of n L).

This paper presents a new interior-point algorithm for linear programming where the constraints are all expressed as inequalities. Along with the concept of minimum-weight basis, the algorithm features a novel mechanism for finding search directions. Unlike other interior-point methods which implicity or explicitly involve optimization over ellipsoids for their direction-finding schemes, the one reported here uses boxes. The corresponding subproblems are simple linear programs having closed form solutions. It is shown that the iterates generated by the algorithm converge to an extreme point of the feasible region. When this point is nondegenerate, it is optimal and reached within finitely any steps. The methodology introduced here also gives rise to a polyhedral subdivision of the problem's feasible region and in fact to the entire space of decision variables.

This document proposes a column-eliminating and a lower bound updating techniques for the simplex method for linear programming. A pricing criterion is developed for checking whether or not a dual hyperplane corresponding to a column intersects a simplex containing all of the optimal dual feasible solutions. If the dual hyperplane has no intersection with this simplex, we can eliminate the corresponding column from the constraints. As the simplex method iterates, the working constraint matrix eventually eliminates all columns except those that are in at least one optimal basis. Keywords: Algorithms; Ellipsoid; Karmarkar method.

We present a problem data set for stochastic programming, and associated real world applications. The problem descriptions were collected from the literature, with emphasis on variety of problem structure and application. Each problem has a short description, mathematical problem statement, and notational reconciliation to a standard problem format. In addition, most problems have one or more corresponding data files in SMPS1 format.

We present a problem data set for stochastic programming, and associated real world applications. The problem descriptions were collected from the literature, with emphasis on variety of problem structure and application. Each problem has a short description, mathematical problem statement, and notational reconciliation to a standard problem format. In addition, most problems have one or more corresponding data files in SMPS1 format.

The existence of neutral /indeterminacy degrees reflects the more practical aspects of decision-making scenarios. Thus, this paper has studied the intuitionistic fuzzy multiobjective linear programming problems (IFMOLPPs) under neutrosophic uncertainty. To highlight the degrees of neutrality in IFMOLPPs, we have investigated the neutrosophic optimization techniques with intuitionistic fuzzy parameters. The marginal evaluation of each objective is determined by three different membership functions, such as truth, indeterminacy, and falsity membership degrees under the neutrosophic environment. The marginal evaluation of each objective function is elicited by various sorts of membership functions such as linear, exponential, and hyperbolic types of membership functions, which signifies an opportunity for decision-makers to select the desired membership functions. The developed neutrosophic optimization technique is implemented on existing numerical problems that reveal the validity and applicability of the proposed methods. A comparative study is also presented with other approaches. At last, conclusions and future research directions are addressed based on the proposed work.

By leveraging advanced techniques and models, Operations Research (OR) provides critical insights and strategic interventions across multiple domains, including transportation, communication, project management, and supply chain optimization. The field's multifaceted approach continues to drive efficiency and innovation in numerous industries. This paper thoroughly evaluates numerous approaches and methods researchers employ to model and investigate problems. Our objective is to bridge gaps in existing literature by examining recent advancements in this field. Network flow problems encompass the shortest path, maximal cost flow, and minimal cost flow problems. These critical elements are essential for understanding transportation dynamics, communication, and resource allocation networks. Furthermore, we explore real-life scenarios where these network flow problems arise, shedding light on their practical significance.

By leveraging advanced techniques and models, Operations Research (OR) provides critical insights and strategic interventions across multiple domains, including transportation, communication, project management, and supply chain optimization. The field's multifaceted approach continues to drive efficiency and innovation in numerous industries. This paper thoroughly evaluates numerous approaches and methods researchers employ to model and investigate problems. Our objective is to bridge gaps in existing literature by examining recent advancements in this field. Network flow problems encompass the shortest path, maximal cost flow, and minimal cost flow problems. These critical elements are essential for understanding transportation dynamics, communication, and resource allocation networks. Furthermore, we explore real-life scenarios where these network flow problems arise, shedding light on their practical significance.

The U.S. Navy Bureau of Medicine and Surgery allocates funding for obstetric staffing resources such as doctors, nurses, and midwives. Furthermore, these resources operate within a fixed number of labor/delivery and postpartum rooms, thereby establishing a theoretical maximum capacity of delivery volume. This study identifies the expected delivery volume created by the facility capacity of four major naval military treatment facilities (MTF) within the United States. Based on the calculated volume, this thesis utilizes a linear programming model to determine the optimum mix of doctors, nurses, and midwives to achieve the target delivery numbers. This is achieved while concurrently incorporating all relevant constraints within military medical treatment facilities. As a result, the model allows hospitals to meet target delivery volumes while simultaneously utilizing their allocated resources in the most effective manner. Additionally, the model can accommodate changes in the inputs and constraints and can be used to provide support for similar resource allocation decision problems.

The U.S. Navy Bureau of Medicine and Surgery allocates funding for obstetric staffing resources such as doctors, nurses, and midwives. Furthermore, these resources operate within a fixed number of labor/delivery and postpartum rooms, thereby establishing a theoretical maximum capacity of delivery volume. This study identifies the expected delivery volume created by the facility capacity of four major naval military treatment facilities (MTF) within the United States. Based on the calculated volume, this thesis utilizes a linear programming model to determine the optimum mix of doctors, nurses, and midwives to achieve the target delivery numbers. This is achieved while concurrently incorporating all relevant constraints within military medical treatment facilities. As a result, the model allows hospitals to meet target delivery volumes while simultaneously utilizing their allocated resources in the most effective manner. Additionally, the model can accommodate changes in the inputs and constraints and can be used to provide support for similar resource allocation decision problems.

The U.S. Navy Bureau of Medicine and Surgery allocates funding for obstetric staffing resources such as doctors, nurses, and midwives. Furthermore, these resources operate within a fixed number of labor/delivery and postpartum rooms, thereby establishing a theoretical maximum capacity of delivery volume. This study identifies the expected delivery volume created by the facility capacity of four major naval military treatment facilities (MTF) within the United States. Based on the calculated volume, this thesis utilizes a linear programming model to determine the optimum mix of doctors, nurses, and midwives to achieve the target delivery numbers. This is achieved while concurrently incorporating all relevant constraints within military medical treatment facilities. As a result, the model allows hospitals to meet target delivery volumes while simultaneously utilizing their allocated resources in the most effective manner. Additionally, the model can accommodate changes in the inputs and constraints and can be used to provide support for similar resource allocation decision problems.

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Planners in multicounty rural areas can use the Rural Development, Activity Analysis Planning (RDAAP) model to try to influence the optimal growth of their areas among different general economic goals. The model implies that best industries for rural areas have: high proportion of imported inputs; low transportation costs; high value added/output ratio and value added/labor ratio; low percentage of labor skill that is most scarce; and low capital/output ratio. The model likewise suggests that conversion of agricultural land from the land use pattern of lower income farms to that of higher income farms is always desirable in a region. The linear programming model was specifically applied, in this report, to a region of northwest Arkansas comprising Benton, Madison, and Washington Counties (BMW region). Alternative objectives included the following: regional balance-of-trade surplus, regional balance-of-payments surplus, gross regional product, local value added, and a regional rate-of-return index. Data were obtained for 1960-70 to compare growth of employment by industry in the model with the actual growth in the area. Because a portion of the BMW region has recently been designated a Standard Metropolitan Statistical Area (SMSA), this study can provide a vehicle for examining how an area might be developed optimally from a rural to a more urban status. (AH)

Planners in multicounty rural areas can use the Rural Development, Activity Analysis Planning (RDAAP) model to try to influence the optimal growth of their areas among different general economic goals. The model implies that best industries for rural areas have: high proportion of imported inputs; low transportation costs; high value added/output ratio and value added/labor ratio; low percentage of labor skill that is most scarce; and low capital/output ratio. The model likewise suggests that conversion of agricultural land from the land use pattern of lower income farms to that of higher income farms is always desirable in a region. The linear programming model was specifically applied, in this report, to a region of northwest Arkansas comprising Benton, Madison, and Washington Counties (BMW region). Alternative objectives included the following: regional balance-of-trade surplus, regional balance-of-payments surplus, gross regional product, local value added, and a regional rate-of-return index. Data were obtained for 1960-70 to compare growth of employment by industry in the model with the actual growth in the area. Because a portion of the BMW region has recently been designated a Standard Metropolitan Statistical Area (SMSA), this study can provide a vehicle for examining how an area might be developed optimally from a rural to a more urban status. (AH)

Planners in multicounty rural areas can use the Rural Development, Activity Analysis Planning (RDAAP) model to try to influence the optimal growth of their areas among different general economic goals. The model implies that best industries for rural areas have: high proportion of imported inputs; low transportation costs; high value added/output ratio and value added/labor ratio; low percentage of labor skill that is most scarce; and low capital/output ratio. The model likewise suggests that conversion of agricultural land from the land use pattern of lower income farms to that of higher income farms is always desirable in a region. The linear programming model was specifically applied, in this report, to a region of northwest Arkansas comprising Benton, Madison, and Washington Counties (BMW region). Alternative objectives included the following: regional balance-of-trade surplus, regional balance-of-payments surplus, gross regional product, local value added, and a regional rate-of-return index. Data were obtained for 1960-70 to compare growth of employment by industry in the model with the actual growth in the area. Because a portion of the BMW region has recently been designated a Standard Metropolitan Statistical Area (SMSA), this study can provide a vehicle for examining how an area might be developed optimally from a rural to a more urban status. (AH)

Experience with solving a 12,753,313 variable linear program is described. This problem is the linear programming relaxation of a set partitioning problem arising from an airline crew scheduling application. A scheme is described that requires successive solutions of small subproblems, yielding a procedure that has little growth in solution time in terms of the number of variables. Experience using the simplex method as implemented in CPLEX, an interior point method as implemented in OB1, and a hybrid interior point/simplex approach is reported. The resulting procedure illustrates the power of an interior point/simplex combination for solving very large-scale linear programs.

Given a point sufficiently close to a nondegenerate basic feasible solution x* of a linear program, we show how to generate a sequence {p-superscript-k} that converges to the 0-1 vector sign(x*) at a Q-cubic rate. This extremely fast convergence enables us to determine, with a high degree of certainty, which variables will be zero and which will be nonzero at optimality and then construct x* from this information.

This article considers the problem of bounding the expected value of a linear program (LP) containing random coefficients, with applications to solving two-stage stochastic programs. An upper bound for minimizations is derived from a restriction of an equivalent, penalty-based formulation of the primal stochastic LP, and a lower bound is obtained from a restriction of a reformulation of the dual. These restricted-recourse bounds are more general and more easily computed than most other bounds because random coefficients may appear anywhere in the LP, neither independence nor boundedness of the coefficients is needed, and the bound is computed by solving a single LP or nonlinear program. Analytical examples demonstrate that the new bounds can be stronger than complementary Jensen bounds. (An upper bound is complementary to a lower bound, and vice versa). In computational work, the authors apply the bounds to a two-stage stochastic program for semiconductor manufacturing with uncertain demand and production rates.

In this report the author presents a heuristic for constructing surrogate constraints to be used for the solution of the linear zero-one integer problem. Using the heuristic the author was able to build surrogate constraints with strength comparable to the dual multiplier surrogate in one-tenth the time. (Author)

A crucial aspect of the Box Method for linear programming is the finding of a minimum-weight basis corresponding to a given interior feasible point. This subproblem leads to the formation of the Box Problem, a special linear program having a closed form solution which provides the search direction at the current iteration. Finding a minimum-weight basis is a matroidal(or, combinatorial) optimization problem that can be handled by a greedy algorithm. This paper suggests a way of efficiently solving the minimum-weight basis problem in cases where the (primal, standard form) linear program contains explicitly bounded variables. It is shown that the main part of the task requires almost no more computational effort or storage space than does a problem of the same size without upper bounded variables. While this result is believed to be valuable in its own right, there is additional benefit to be gained in applications where the finding of a minimum-weight basis (for a linear program without explicit upper bounds on its variables) is done by a special greedy algorithm. Such is the case with minimum-cost network flow problems which will be discussed in Part III of this series.

This paper discusses the problems involved in using linear programming as a subroutine of a larger routine. Proposals are made for eliminating the tolerance selection problem, and for improving the accuracy of inversions. Sample programs are given in FORTRAN IV and in ALGOL.

A linear programming system which is the modified simplex procedure with the product form of inverse is discussed. It is designed to solve the classical linear inequalities problem and most of its variations on the IBM 704.

A linear program is considered with several objective functions. The traditional approach has been either to 'trade off' by weighting each function, or if a 'trade off' vector cannot be provided to ignore all but the most significant. The author is interested in classes of programs whose members possess some common characteristics. Examples are sequences of production, refining, inventory problems over time at one installation. If sufficient conditions exist an estimate of a 'trade-off' vector can be made. This estimate improves over the sequence. A set X* exists which contains the solutions obtained by optimizing with respect to all nonnegative combinations of objective functions. A decision maker is not indifferent to these solutions but can characterize preferred solutions. A method is presented whereby he can direct a finite sequence of solutions over X* towards a preferred solution. As the estimate of the 'trade-off' vector improves, the expected length of the sequence diminishes, and the efficiency of solution increases.

The linear programs considered here are of the form: Maximize (c,x) (LP) subject to a = or A x = or b where A is of full row rank, and (LP) is feasible with bounded optimal solutions. The main result is an explicit representation of the general optimal solution of (LP), in terms of a generalized inverse of A. This explicit solution of (LP) - explicit in the sense that A superscript (-1) b is an explicit solution of A x = b - has obvious theoretical (and possibly computational) advantages over the well known iterative methods of linear programming. The results are illustrated by a simple example, and extensions to general linear programs are discussed.