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When all the fire companies in a region are engaged in fighting fires, protection against a future fire is considerably reduced. It is standard practice in many urban fire departments to protect the exposed region by relocating outside fire companies temporarily to some of the vacant houses. In New York City, situations requiring such relocations arise ten times a day on the average. The Fire Department of the City of New York (FDNY) currently makes its relocations according to a system of preplanned moves. This system was designed at a time when alarm rates were low and is based on the assumption that only one fire is in progress at a time. Because of the high alarm rates currently being experienced in parts of New York City, this assumption is no longer valid, and the preplanned relocation system breaks down at the times when it is needed most. This paper describes a computer-based method for determining relocations that overcomes the deficiencies of the existing method by utilizing the computer's ability to (1) store up-to-date information about the status of all fires in progress and the location and activity of all fire companies, (2) generate and compare many alternative relocation plans quickly. The method, which will become part of the FDNY's real-time Management Information and Control System (MICS), is designed to be fast and to require little computer memory. After giving some background of the problem and the objectives of relocation, we give the problem a mathematical programming formulation and then describe the heuristic algorithm to be used for generating relocations in the MICS. The remainder of the paper is devoted to a discussion of an example illustrating how the algorithm works, a rigorous test of the algorithm using a computer simulation model of Fire-Department operations, and a description of the current use of... [ABSTRACT FROM AUTHOR]

The purpose of rids paper is to develop a constructive algorithm for generating the admissible boundary of a convex polytope, when the polytope is the convex hull of a given (finite) set of points, as well as to mention some related results. One application is to the sensitivity of decisions to changes in the prior distribution for problems in statistical decision theory. [ABSTRACT FROM AUTHOR]

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 considers fluid analogues for the standard linear programming problem and for a separable nonlinear programming problem. In the former case the usual duality results are demonstrated using the principle of minimum potential energy. In addition by examining the dynamics of the system a new method, referred to as the R-method, is derived for solving linear programmes, although it is not demonstrated that this has any computational advantages over the standard simplex method, except that degeneracy causes no problems. In the nonlinear case the weak Lagrangian principle is derived. The purpose of the paper is to demonstrate that analogue methods, while being impracticable as a physical method of solving optimisation problems, may give some insight into computational algorithms via dual energy concepts and/or dynamic behaviour. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to furnish a computational scheme for a class of programming problems with nonlinear constraints. The algorithm is sequential in nature, producing a sequence of feasible solutions whose limit points are optimal solutions of the original problem. Further, with a view to accelerating convergence we suggest two variants of the solution procedure. [ABSTRACT FROM AUTHOR]

A mean-variance portfolio selection model suitable for the small investor is formulated as a sequence of quadratic integer programming problems. The special structure of these quadratic problems is exploited in a partial enumeration algorithm which uses cutting planes to accelerate convergence. Computational experience is reported on problems ranging in size from fifteen to fifty variables. [ABSTRACT FROM AUTHOR]

This part of the paper introduces some possible implementations of Self-Scaling Variable Metric algorithms based on the theory presented in Part I. These implementations are analyzed theoretically and discussed qualitatively. A special class of SSVM algorithms is introduced, which has the additional property of being invariant under scaling of the objective function or of the variables. Experimental results are provided for a particular case of this class. This case has been tested in comparison to the DFP algorithm on a variety of functions with up to 50 variables. The results indicate that the new method has substantial advantage for functions with a large number of variables. [ABSTRACT FROM AUTHOR]

This paper presents an algorithm for solving resource-constrained network scheduling problems, a general class of problems that includes the classical job-shop-scheduling problem. It uses Lagrange multipliers to dualize the resource constraints, forming a Lagrangian problem in which the network constraints appear explicitly, while the resource constraints appear only in the Lagrangian function. Because the network constraints do not interact among jobs, the problem of minimizing the Lagrangian decomposes into a subproblem for each job. Algorithms are presented for solving these subproblems. Minimizing the Lagrangian with fixed multiplier values yields a lower bound on the cost of an optimal solution to the scheduling problem. The paper gives procedures for adjusting the multipliers iteratively to obtain strong bounds, and it develops a branch-and-bound algorithm that uses these bounds in the solution of the scheduling problem. Computational experience with this algorithm is discussed. [ABSTRACT FROM AUTHOR]

This paper examines Markovian decision processes in which the transition probabilities corresponding to alternative decisions are not known with certainty. The processes are assumed to be finite-state, discrete-time, and stationary. The rewards are time discounted. Both a game-theoretic and the Bayesian formulation are considered. In the game-theoretic formulation, variants of a policy-iteration algorithm are provided for both the max-min and the max-max cases. An implicit enumeration algorithm is discussed for the Bayesian formulation where upper and lower bounds on the total expected discounted return are provided by the max-max and max-rain optimal policies. Finally, the paper discusses asymptotically Bayes-optimal policies. [ABSTRACT FROM AUTHOR]

This paper presents a model for designing multiactivity, multifacility systems. Given the maximum number of facilities and their possible locations, the problem is to determine which facilities to include in the system and which activities should be carried on at each facility in order to minimize the cost of the system. A branch-and-bound algorithm for solving the problem is given; the subproblems solved by the algorithm are tractable integer linear programs. The model can be used in the design of such diverse systems as neighborhood health centers, schools, and multiproduct inventory distribution systems. [ABSTRACT FROM AUTHOR]

This paper presents new types of cutting-plane algorithms for integer-constrained optimization problems. The underlying idea of the method of enumerative cuts is to make use of an enumeration procedure in order to construct cutting planes that can be made arbitrarily deep. In recent publications, BALLAS and YOUNG have established in somewhat different contexts that valid cutting planes can be generated from the intersection of (basic) rays with adequately constructed hyperspheres. This paper introduces a family of polyhedra to replace the sphere; these polyhedra are not tangential to the hypersphere and produce cutting planes with particular characteristics that are discussed. Constructive procedures are sketched for generating step-by-step cutting planes that become deeper and deeper at every step, until the deepest cut is obtained. This construction often relies on a partial enumeration scheme of the set of integer solutions. [ABSTRACT FROM AUTHOR]

This paper treats the problem of project (or machine) scheduling with resource constraints to achieve minimum total duration time as a disjunctive graph. The prospective advantage of this approach is the elimination of the need to consider individual time periods over the program horizon; a feasibility check determines whether the resource constraints can be met by any particular network representation of the project. The paper gives an algorithm that uses partial enumeration for what is essentially a mixed integer program. The algorithm employs a maximum-flow computation as a check for feasibility with respect to available resources. [ABSTRACT FROM AUTHOR]

The generalized network problem and the closely related restricted dyadic problem occur frequently in applications of linear programming. Although they are next in order of computational complexity after pure network or distribution problems, the jump in degree of difficulty is such that, in the most general problem, there exist no algorithms comparable in speed to those for pure networks. In this paper we characterize the properties of a special class of generalized network problems that permit a dual feasible basic solution to be determined in one 'pass' through the network. In particular, this class includes the class of pure network problems for which no such procedure has previously existed. Our algorithm also makes it possible to apply LEMKE'S dual method and the poly-ω technique of CHARNES AND COOPER in an efficient manner to solve capacitated (pure) network problems. [ABSTRACT FROM AUTHOR]

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

This paper develops planning-horizon theorems and algorithms for determining optimal production in period 1 when there are linear costs associated with overtime and inventory, and demand is deterministic. If overtime in each period is unrestricted and no shortages are permitted, then it is easily shown that the firm will have to know future demand for at most the next k* periods, where k* is determined by the ratio of the wage differential between overtime and regular time to the per unit holding cost per period. Using the k* concept, the paper determines planning horizons when both storage and backlogging of demand are allowed. If overtime constraints are imposed, a planning algorithm is proposed that requires less information on future demand than a linear programming model. Economic implications of this approach for decision-making and information processing in firms are discussed. [ABSTRACT FROM AUTHOR]

This paper gives a simple algorithm for finding optimal allocations subject to a hierarchy of limits when the loss function is separable strictly convex and the resources function is linear. Its applications to capital budgeting and multistage sampling are pointed out. The allocation problem considered by Srikantan [Opns. Res. 11,265-273 (1963)] is a special case of the problem considered in this paper. [ABSTRACT FROM AUTHOR]

A mathematical programming problem P: minimize z=d[supT]x+e[supT]u+f[supT]v subject to Ax+Bu+Cv≥g, x, u, v≥0, u[supT]v=0, is called a complementary programming (CP) problem. The condition u[supT]v=0 distinguishes CP problems from ordinary LP problems. A variety of nonlinear programming problems can be formulated in this form, including absolute-value programming problems, 0-1 mixed-integer-programming problems, quadratic-programming problems, and so forth. This paper proposes a branch-and-bound algorithm for CP problems and reports some computational results. [ABSTRACT FROM AUTHOR]

Heuristic algorithms have frequently been developed in the attempt to solve combinatorial problems such as those that frequently exist in plant-layout planning, sequencing, scheduling, and routing situations. In these algorithms there has been no basis available to evaluate the progress of the heuristic results in relation to an optimum. This paper outlines and provides examples of the application of extreme value distributions and, more particularly, the Weibull distribution as mechanisms for estimating optimum limits. Given such estimates, the paper further illustrates the development of decision rules applicable to the decision of continuing or halting the heuristic analysis. [ABSTRACT FROM AUTHOR]

This paper studies the optimal operation of an M/G/1 queuing system with removable server and the following cost structure: a holding cost per customer in the system per unit time, a cost per unit time of keeping the server running, and fixed charges for turning the server on or off. The server can be turned on at arrival epochs or off at service-completion epochs. The paper characterizes an optimal policy for the infinite-horizon discounted problem, offers an optimality proof, and presents a computational algorithm. [ABSTRACT FROM AUTHOR]

This paper combines the theory of generalized Lagrange multipliers with a reformulation of the integer-programming problem due to group theory. The use of multipliers enhances the algorithmic efficiency of group theory in a variety of ways. One particular application is the approximation of generalized Lagrange multipliers by generalized linear programming as suggested by BROOKS AND GEOFFRION. This procedure is shown to be closely related to the cutting-plane method of GOMORY. [ABSTRACT FROM AUTHOR]

This paper proposes a new class of cutting planes for integer programming. A typical member of the class is generated as follows. Let X be the feasible set, and ... the optimal (noninteger) solution to the linear program associated with an integer program in n-space. Consider a unit hypercube containing 2, whose vertices are integer, and the hypersphere circumscribing the cube. This hypersphere is intersected in n independent points by the n hairlines originating at ... and containing the n edges of X adjacent to (if ... is degenerate, X is replaced by X' ⊃ X having exactly n edges adjacent to ...). The hyperplane through these n points of intersection defines a valid cut, the (spherical) intersection cut. The paper gives a simple formula for finding the equation of the hyperplane, discusses some ways of strengthening the cut, proposes an algorithm, and gives a finiteness proof. A straightforward extension of these geometric ideas yields an analogous (cylindrical) intersection cut for the mixed-integer case. A discussion of relations to other work (Young, Gomory) is followed by the introduction of some additional intersection cuts obtained from hypersurfaces other than the sphere or the cylinder. The message of this paper lies, not so much in the particular cuts that it proposes, as in the basically new approach to integer programming that these cuts typify. This approach is geometrically motivated and uses the 'local' properties (in the integer-programming sense) of the feasible set. [ABSTRACT FROM AUTHOR]

This paper proposes a general method of solution for problems, involving the allocation of jobs to machines and the sequencing of jobs on machines, in which a number of jobs, each consisting of a number of operations that may be performed on a given set of machines, is to be completed in some optimal fashion. Considering constraints on machine availability, job completion dates, and zoning, the paper demonstrates the underlying relation of a wide variety of machine-job problems with reference to their graphical representation, states a branch-and-bound method of solution, solves an illustrative example, and gives the mode of application to specific problems (for several of these the method reduces to methods published elsewhere). The algorithm represents not only a general framework of theoretical interest, but also a practical approach to certain specific problems. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present an algorithm for solving the quadratic binary programming problem. Although a problem with this structure may arise in many situations, it is particularly common in capital budgeting when a decision-maker is confronted with a set of investment proposals from which he must select a portfolio. If returns of proposals are intercorrelated random variables and if the decision-maker uses as his criterion for selection the mean μ and variance σ[sups] of portfolio returns, his decision requires prior identification of the (μ, σ²) efficient set. The algorithm developed to solve the problem and hence necessary to generate the efficient set is based on the concept of implicit enumeration recently introduced by EGON BALAS for solution of the binary linear programming problem. [ABSTRACT FROM AUTHOR]

One formulation of the machine sequencing problem is that of finding a minimaximal path in a disjunctive graph. This paper describes an implicit enumeration procedure that solves the problem by generating a sequence of circuit-free graphs and solving a slightly amended critical-path problem for each graph in the sequence. Each new term of the sequence is generated from an earlier one by complementing one disjunctive arc. The search tree is drastically cut down by the fact that the only disjunctive arcs that have to be considered for being complemented are those on a critical path. An evaluation of these candidates is used to direct the search at each stage, The procedure can start with any feasible schedule (like the one actually used in production, or generated by some heuristics), and gradually improve it. Thus one can possibly stop short of the optimum, with a reasonably 'good' feasible schedule. Storage requirements are limited to the data pertinent to the current node of the search tree. [ABSTRACT FROM AUTHOR]

This paper presents an efficient algorithm for finding a minimum-weight generalized matching in a weighted bipartite graph. Computational evidence is given that indicates that the time required to find a least-cost assignment of n jobs to n workers goes roughly as n[sub2] for 10=

The algorithms of this paper belong to the direct-search or implicit-enumeration type. They compare to the recently proposed algorithm of EFROYM- SON AND RAY, as does the mixed integer algorithm proposed by C. E. LEMKE and the author to that of LAND AND DOIG The general plan of procedure is expected to be equally valid for the capacitated plant-location problem and for transshipment problems with fixed charges; with some of the proposed devices more important for these difficult problems than for the simple plant location problem. Considerable computational experience has been accumulated and is discussed at some length. It suggests that additional work on the construction of adaptive' programs, matching algorithm to data structure, is desirable. [ABSTRACT FROM AUTHOR]

This paper proposes an algorithm for maximizing a linear (unimodal) function over a compact convex set X in n-dimensional Euclidian space. The algorithm is a variant of the cutting-plane method of Cheney and Goldstein and of Kelley. In their method one maximizes the original linear (unimodal) function over successively refined approximations to X. Each approximation is the intersection of finitely many closed half spaces containing X. Any limit point of the sequence of solutions to the approximating problems solves the original problem. The principal new feature of our method is that each successive half space is defined by a hyperplane that supports X at a boundary point (ZOUTENDIJK'S MFD method also has this property). The particular boundary point is chosen as the point where the line drawn from a fixed interior point of X to the solution (exterior to X) to the most recent approximating problem pierces the boundary of X. Our method is applicable where X is described as the set on which finitely many unimodal functions are nonnegative. The cutting-plane method is applicable only under the more restrictive assumption that the above functions are concave. Our method is formulated so as to be applicable also to unimodal integer programs. [ABSTRACT FROM AUTHOR]

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

This paper develops a methodology to evaluate various aspects of logistics supply support of space bases. It is assumed that there exists at the space base a schedule of operations that reflects the day-to-day living, build-up, and scientific experimental activities that are to be carried on. These activities, in turn, set a series of demands or requirements for products over a time spectrum. The supply system must deliver products so as to meet the amounts and times of the product requirements. Each product or module has an earliest and latest time by which it must be delivered. A mathematical model is developed that plans a series of trips, the dates at which each is to be sent, and the composition of the cargoes on each trip that satisfy the series of requirements over a time spectrum imposed by the activities at the space base. The series of trips are an expression of an efficient plan that simultaneously considers demands for different products at different future times and observes the various constraints of the system (e.g., cargo capacity of spaceships). The mathematical formulation of this scheduling problem is a simple non- linear discrete programming problem. An algorithm has been developed for its solution, and a description thereof is presented. The model is illustrated by showing how it would supply a long-term lunar base. Various uses of the model are described. [ABSTRACT FROM AUTHOR]

Following a line of approach recently applied to the 0-1 integer programming problem with some success by EGON BALAS, the algorithm of this paper is based upon an underlying tree-search structure upon which a series of tests is superimposed to exclude large portions of the tree of all possible 0-1 solutions from examination. In our method, the specific design of the enumeration and tests, supplemented by the use of a special type of constraint called a 'surrogate constraint,' results in an algorithm that appears to be quite efficient in relation to other algorithms currently available for solving the 0-1 integer programming problem. Early indications of efficiency must, however, be regarded as suggestive rather than conclusive, due to the limited range and size of problems so far examined. Following the analytical development of the method, three example problems are solved in detail with the Multiphase-Dual Algorithm to illustrate various aspects of its application. An extension of the algorithm to the general integer programming problem in bounded variables is briefly sketched in a concluding section. [ABSTRACT FROM AUTHOR]

This paper summarizes experimental results from applying several computational methods for solving the classic three-machine scheduling model. The basic mathematical problem is to select an optimal permutation of a items, where the objective function employed is the total amount of processing time elapsing for the completion of all n items on three machines. The methods tested are integer linear programming, ordinary linear programming with answers rounded to integers, a heuristic algorithm, and random sampling. Throughout the experimentation n=6. A dual integer programming code is tested on six trial problems. Consideration is given to studying the phenomenon of input-form sensitivity. The results are not encouraging and confirm previous experimental evidence as to the strong effect of varying the input form. However, the addition of a powerful bound on the objective function has a significant beneficial effect on the convergence of the dual algorithm. A sampling study of the statistical characteristics of this model is made with 100 sets of data randomly generated. Frequency distributions of minimum and mean total processing time are tabulated and analyzed. On this same set of problems, a linear programming approach is examined for effectiveness in producing nearly optimal schedules. Such an approach tends to give answers that are better than would be obtained by a single permutation drawn at random, but the LP rounded solutions average about an 11 per cent increase over the optimal processing time. A heuristic algorithm based on Johnson's method is tested on 20 of the 100 eases and gives excellent results. Finally attention is turned to the method of randomly sampling the permutations, possibly applying a simple heuristic gradient method to improve each sampled schedule. A feature of this method is that elementary probability theory can be applied to yield corresponding statements about the accuracy guaranteed by the approach. [ABSTRACT FROM AUTHOR]

Because the payload that an airplane can carry is a function of the longest segment of its route and differs from route to routes the usual algorithms of network flow theory will not yield the optimum flow of payload. In this paper two network algorithms are described. They determine the route of maximum payload flow per hour of flight time and the maximum steady- state payload flow through a network with capacity constraints on the nodes,. [ABSTRACT FROM AUTHOR]

The 'explosion' problem is the problem of determining the required components from knowledge of demand for finished products. The 'netting' problem IS the problem of determining the net requirements of these components, for purposes of production or procurement from outside the firm, based on available inventory. This paper presents a simple algorithm which yields the 'netted explosion. 'The algorithm is suitable for digital computation in case of large inventories. [ABSTRACT FROM AUTHOR]

This paper describes an algorithm for a ship scheduling problem, obtained from a Swedish ship-owning company. The algorithm uses the Dantzig-Wolfe decomposition method for linear programming. The subprograms are simple network flow problems that are solved by dynamic programming. The master program in the decomposition algorithm is an LP problem with only zero-one elements in the matrix and the right-hand side. Integer solutions are not guaranteed, but generation and solution of a large number of problems indicates that the frequency of fractional solutions is as small as 1-2 per cent. Problems with about 40 ships and 50 cargoes are solved in about 2.5 minutes on an IBM 7090. In order to resolve the fractional cases, some integer programming experiments have been made. The results will be reported in a forthcoming paper. [ABSTRACT FROM AUTHOR]

This paper studies the close relation between the Gass-Saaty parametric programming algorithm and the `primal-dual' procedures recently exploited by DANTZIG, FORD, AND FULKERSON lit is shown that the two procedures are equivalent The possibility of eliminating the two-phase character of the simplex method using these techniques is discussed Finally, the application of the techniques to problems with special structure is considered [ABSTRACT FROM AUTHOR]

This paper is concerned with the following problem in network synthesis. Suppose that we are given a network with real-valued capacities on each arc. There is a cost associated with each arc which is proportional to the magnitude of the arc capacity. We wish to determine new arc capacities such that the capacity of each cut in the new network differs from the capacity of the corresponding cut in the original network by a specified constant and such that the total cost is minimized. This problem is shown to be equivalent to solving a minimum cost flow problem on a related network. [ABSTRACT FROM AUTHOR]

Randomly-generated linear programming problems of three different types and five different sizes were solved by the criss-cross method and by the simplex method. One hundred problems of each type and size were solved, and the results are overwhelmingly favorable to the criss-cross method. An improvement to the criss-cross method used in these tests is given, and the extension of the results of the paper to variations of the criss-cross and simplex methods is discussed. [ABSTRACT FROM AUTHOR]

The paper considers a class of optimization problems. The problems are linear programming problems: maximize ex subject to Ax = b with the additional constraint that x must also be an extreme point of a second convex polyhedron Dx = d. X ≥ 0. A cutting-plane algorithm for solving such problems is presented. Two numerical examples are also included. [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.

This paper presents a model for determining how a central government can most efficiently allocate resources among other levels of government. The model explicitly includes the fact that lower levels of government can make independent decisions once they have been given resources by the central government. A key feature of the model is the mathematical formulation of the central government's objective of distributing resources efficiently, while at the same time being as fair as possible to all those receiving allocations. An algorithm for solving the model is presented along with a numerical example. [ABSTRACT FROM AUTHOR]

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]

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]

This paper extends the results of "Duality in Discrete Programming" [1] to the case of quadratic objective functions. The paper is, however, self-contained. A pair of symmetric dual quadratic programs is generalized by constraining some of the variables to belong to arbitrary sets of real numbers. Quadratic all-integer and mixed-integer programs are special cases of these problems. The resulting primal problem is shown, subject to a qualification, to have an optimal solution if and only if the dual has one, and in this case the values of their respective objective functions are equal. The dual of a mixed-integer quadratic program can be formulated as a minimax problem whose quadratic objective function is linear in the integer-constrained variables, and whose linear constraint set does not contain the latter. Based on this approach an algorithm is developed for solving integer and mixed-integer quadratic programs. [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 presents an algorithm for solving a linear integer programming problem and describes the theoretical foundations for a new approach to integer programming. According to the authors, the approach may be described as an extension of the methods using enumeration, but by using a new, more powerful approach based on population statistics. The optimal solution to the integer programming problem in this paper is viewed as one of selecting the optimal function among a certain class of functions which map elements of one set into another. With this method in mind, the authors intend to view the different problems by characterizing the statistical structure of the associated finite class of maps.

The usefulness of integer programming as a tool of capital budgeting hinges on the development of an efficient solution technique. An algorithm based on partial enumeration has been developed by E. L. Lawler and M. D. Bell for solving integer linear programs with 0-1 decision variables; however their algorithm is not general enough to deal with all problems in which the objective function is quadratic. This paper extends Lawler and Bell's method so that it can be generally applied to integer quadratic programs. The new algorithm is illustrated by examples from capital budgeting. [ABSTRACT FROM AUTHOR]

The literature is replete with analyses of minimum cost flows in networks for which the cost of shipping from node to node is a linear function. However, the linear cost assumption is often not realistic. Situations in which there is a set-up charge, discounting, or efficiencies of scale give rise to concave functions. Although concave functions can be minimized by an exhaustive search of all the extreme points of the convex feasible region, such an approach is impractical for all but the simplest of problems. In this paper some theorems are developed which explicitly characterize the extreme points for certain single commodity networks. By exploiting this characterization algorithms are developed that determine the minimum concave cost solution for networks with a single source and a single destination, for acyclic single source multiple destination networks, and for acyclic single destination multiple source networks. An interesting theorem then establishes that for either single source or single destination networks the multi-commodity case can be reduced to the single commodity case. [ABSTRACT FROM AUTHOR]

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