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This paper examines properties and interrelations of several concepts of generalized concavity. It shows that the class of functions that are both ‘generalized concave’ and ‘generalized convex’ is closely related to the class of monotone functions of a single variable. After excluding a certain small class of exceptions, the paper shows that, for arbitrary (perhaps not differentiable) functions, concave implies pseudoconcave, pseudoconcave implies strictly quasiconcave, and strictly quasiconcave implies quasiconcave. Several results characterizing the extreme values of generalized concave functions are given. [ABSTRACT FROM AUTHOR]

This paper investigates the optimal allocation of the marketing budget within the marketing-mix decision variables so that sales (or profit) is maximized in a planning horizon. Since the influence of marketing mix variables upon sales are, in reality, nonlinear and interactive, a geometric programming algorithm is used that solves this problem. A procedure to estimate a functional of sales on the marketing mix and environmental variables utilizing the experienced judgments of the firm's executives and the raw data is provided. The derived functional is later optimized by the Geometric Programming algorithm under a constraint set consisting of budget and strategy restrictions imposed by a firm's marketing environment, and conditions under which the optimal solution is either local or global are identified. An empirical application for a large midwestern brewery is provided which utilizes and illustrates both the estimation an optimization procedures. [ABSTRACT FROM AUTHOR]

This paper deals with systems (such as electronic devices and corporations) that are designed to accomplish multiple goals and whose internal structures are characterized by networks of interacting component parts. Each component of such a system is typically associated with a probability of failure, while each system goal is associated with a reward value. Failure of any network component may ultimately result in the inability to achieve one or more goals and force the forfeiture of the associated rewards. A basic problem in component-system design is to determine how the expectation of total reward may be maximized through the cost-limited acquisition of redundant (backup) components. This paper provides a formal statement of this redundancy-optimization problem and argues that the problem may not be solved easily by standard linear or integer programming techniques. It introduces an algorithm to solve this problem, proves its convergence, and presents computational results taken from a battery of test problems and an algorithm-efficiency study based on these tests. [ABSTRACT FROM AUTHOR]

This paper treats the optimal location of nuclear-power plants in the Pacific Northwest with respect to capital-construction, operating, and transmission costs. It presents a mathematical formulation of this two-product (peak and energy power) plant-location problem, using the simplex method in conjunction with a branch-and-bound process, and indicates a check procedure using Vogel's approximation method in conjunction with an algorithm for solving the transportation problem and a branch-and-bound process. The paper gives results for demand-growth rates of 4, 5, 6 and 7 percent and alternative designs and their costs. It also discusses the stability of the optimal solution. [ABSTRACT FROM AUTHOR]

This paper develops a new cutting plane for primal integer programming. This cut, when it exists and is used as pivot row, has the property that the minimum of the simplex evaluators of the next tableau is strictly larger (by an integer) than the minimum for the current tableau. The paper gives a sufficient condition in the form of a linear program for the existence of such a cut, and proposes an algorithm for generating such cuts. It also presents two additional sufficient conditions. The process of cut generation is then imbedded into a convergent primal algorithm to be used in an attempt to overcome the proof-of-optimality problem experienced with the primal algorithm. [ABSTRACT FROM AUTHOR]

This paper presents an algorithm for finding the K loopless paths that have the shortest lengths from one node to another node in a network. The significance of the new algorithm is that its computational upper bound increases only linearly with the value of K. Consequently, in general, the new algorithm is extremely efficient as compared with the algorithms proposed by Bock, Kantner, and Haynes [2], Pollack [7], [8], Clarke, Krikorian, and Rausan [3], Sakarovitch [9] and others. This paper first reviews the algorithms presently available for finding the K shortest loopless paths in terms of the computational effort and memory addresses they require. This is followed by the presentation of the new algorithm and its justification. Finally, the efficiency of the new algorithm is examined and compared with that of other algorithms. [ABSTRACT FROM AUTHOR]

This paper analyzes the problem of balancing resource capacity utilization against the costs of user delay or inventory investment. Instead of formulating algorithms to solve more classical versions of the problem, this paper describes ways that organizations can drop the cost curve prior to applying an optimizing algorithm. This is done by classifying the techniques and giving examples from organizations which have faced serious smoothing problems. Other organizations can improve their coat performance by adopting some of these techniques. [ABSTRACT FROM AUTHOR]

In an earlier paper [20] combinatorial programming procedures were presented for solving a class of integer programming problems in which all elements are zero or one. By representing the problem elements in a binary computer as bits in a word and employing logical "and" and "or" operations in the problem-solving process, a number of problems involving several hundred integer variables were solved in a matter of seconds. In the present paper a number of improvements in these earlier algorithms are presented, including a new search strategy, methods for reducing the original problem, and mechanisms for feasibility filtering in multi-word problems. With these improvements problem-solving efficiency has been increased in many instances by an order of magnitude. In addition, the present paper contains computational experience obtained in solving problems for the k-best solutions. [ABSTRACT FROM AUTHOR]

A complementary programming problem is a linear programming problem with the additional restriction that xpxq=0 holds for each specified pair (xp, xq). This paper obtains constraints called C-cuts from the restriction xpxq=0, and uses them to facilitate the computation of the branch-and-bound procedure proposed in an earlier paper [Opns. Res. 19, 1523–1528 (1971)]. Some computational results are also reported. [ABSTRACT FROM AUTHOR]

This paper introduces a generalization of the frequently discussed problem of finding the optimum redundancy that maximizes the reliability of a network of components. Past work has restricted consideration to arrangements of redundant components called series-parallel networks. This paper allows a much broader class of arrangements called series-parallelseries networks. It is important to consider such arrangements for realistic situations in which components have more than one failure mode, or the combination of parallel paths introduces a failure probability. A dynamic programming algorithm is used to solve the more general problem for the case in which there are no constraints on the optimum solution. The algorithm is extended to handle multiple constraints using dominance and a variety of elimination methods to reduce the storage required in a compurer implementation of the algorithm. Problems with as many as 15 serial components and three constraints have been solved with reasonable digital computer computation times. [ABSTRACT FROM AUTHOR]

This paper develops a new algorithm for solving the general nonlinear programming problem that melds the flexible simplex search of NELDER AND MEAD with various additional rules to take care of equality and/or inequality constraints. The set of violated inequalities and equalities is lumped into one inequality constraint loosely satisfied during the early progress of the optimization and more closely satisfied during its final stages. To permit this type of search, the method sets up a special tolerance criterion, a function that does not depend on either the values of the objective function or the values of the constraints. The new algorithm has solved successfully a number of problems that have been proposed in the literature as test problems. Finally, to indicate the algorithm's capabilities, the paper describes an example composed of a linear objective function of twenty-four variables subject to fourteen nonlinear equalities and thirty inequalities. [ABSTRACT FROM AUTHOR]

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

This paper considers the static flowshop-scheduling problem with the objective of minimizing, as a cost function, the mean job-completion time. Within the more general framework of combinatorial optimization problems, it defines a heuristic search technique--an approach that has been successful in the past in obtaining near-optimal solutions for problems that could not be solved exactly, either for lack of theory or because of exorbitant computational requirements. The paper presents a two-phase algorithm: The first phase searches among schedules with identical processing orders on all machines; the second refines the schedule by allowing passing. Results of computer study are presented for a large ensemble of pseudorandom problems, and for two particular problems previously cited in the literature. The method is shown to provide solutions that are exceptionally low in cost, and superior to those provided by sampling techniques in the cases for which comparison is possible. Computation time is also discussed and is given in machine-independent terms. [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]

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.

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]

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

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

The 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]

This paper presents an algorithm for the solution of dynamic programming problems requiring the determination of optimal policies for the control of a special class of stochastic processes when the time horizon of the planning period is at infinity. These processes can be mathematically described as discrete time parameter Markov chains with a finite number of states which have been "embedded" in continuous time in the sense that the time between transitions is a random variable whose probability distribution depends only on the states between which the transition takes place. Such processes are called Markov-renewal processes. The Markov processes considered by R. A. Howard in [1] are really two special cases of this somewhat wider class of stochastic processes. In these two special cases, the algorithm of this paper is identical with Howard's. In fact, with only slight modification, Howard's algorithm can be extended to this wider class of stochastic processes. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to illustrate some applications of the functional equation technique of the theory of dynamic programming to a general class of problems arising in the study of networks, particularly those arising in transportation theory. [ABSTRACT FROM AUTHOR]

In this paper, it is shown that the "caterer" problem, a problem in mathematical economics and logistics which has been discussed by Jacobs, Gaddum, Hoffman and Sokolowsky, and Prager, can be reduced to the problem of determining the maximum of the linear form Ln = ∑I=1nv1, subject to a series of constraints of the form V1 ≦ b1, v1 +v2 ≦ b2, v1 + v2 +v3 ≦ b2, ߪ, v1 + v2 + … + vK ≦ bK, v2 + v3 + … + vK+1 ≦ bK+1, …, vb-h+1 + … + Vn ≦ bn, 0 ≦ rI, I = 1, 2, … , n, under an assumption concerning the nonaccumulation of dirty laundry. This maximization problem is solved explicitly, using the functional equation technique of dynamic programming. [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]

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 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]

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

This paper treats the problem of locating a given number of interrelated physical facilities in a single- or multi-story building as an optimization model such that the weighted sum of the distances along orthogonal directions is minimized, and describes a constructive heuristic algorithm for the one-dimensional model. This linear placement algorithm is then extended to higher dimensions for placing uniform-size equipment modules at homogeneous fixed locations. The algorithm accommodates lower-bound constraints for the linear case efficiently and produces near-optimal solutions. Its significance, as demonstrated by comparison with some existing heuristic procedures, is that it has the capability of handling efficiently problems with large numbers of equal-size facilities whose solutions are computationally intractable using the present optimum-producing methods. [ABSTRACT FROM AUTHOR]

This paper presents an algorithm for the global maximization of a convex function subject to linear inequality constraints. It is computationally finite and is designed to converge rapidly on problems in which there are few local optima or the global optimum is significantly better than most of the other local optima. [ABSTRACT FROM AUTHOR]

This paper gives counterexamples to: (1) Ritter's algorithm for the global maximization of a quadratic subject to linear inequality constraints, and (2) Tui's algorithm for the global minimization of a concave function subject to linear inequality constraints. [ABSTRACT FROM AUTHOR]

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

In this paper we apply the concept of generalized Lagrange multipliers, introduced by Everett [4], to the development of an algorithm for a one-period multiproduct production model, where the objective is to maximize profit subject to constraints on aggregate regular time and overtime production. We assume no difference between the cost of idle time and regular time labor, so that the regular time cost is fixed. Overtime cost is variable. The price for each product is constant (independent of quantity sold), and everything produced can be sold. The optimum production schedule (maximum profit) will depend upon revenues, overtime costs, setup times, and productivities. [ABSTRACT FROM AUTHOR]

This paper describes a simple algorithm for the solution of very large sequence problems without the use of a computer. It produces approximate solutions to the n job, m machine sequencing problem where no passing is considered and the criterion is minimum total elapsed time. Up to m-1 sequences may be found. [ABSTRACT FROM AUTHOR]

The general practice in mixed model assembly line balancing is to assign work to stations in a manner that each station has an equal amount of work on a daily or shift basis. This paper shows how a modification to mixed model line balancing algorithms can be used so that the stations are loaded more consistently on a model by model basis as well. In addition, the modification now allows mixed model line balancing to become a plausible tool for hatched assembly line processes. The procedure is general in nature and is applicable to most line balancing algorithms. [ABSTRACT FROM AUTHOR]

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

If the constraint matrix of a linear program has special structure it may be possible to speed computation. Techniques have been developed to take advantages of such special structures as generalized networks, generalized upper bounding, and decomposition. For these matrix structures, it is shown in this paper how to extend the techniques to Zangwill's mathematical programming algorithm, the convex simplex method. [ABSTRACT FROM AUTHOR]

There exists computational experience to the effect that search algorithms of the type introduced by Balas are frequently ineffective because of being tied to a rigid "natural" search origin. In this paper the dynamic relocation of the search origin to possibly more appropriate points is advocated. Details of such a procedure are given for the Simple Plant Location Problem. Computational improvements of, sometimes, a striking nature are presented. [ABSTRACT FROM AUTHOR]

Multi-level fixed-charge problems are mathematical optimization problems in which the separable portion of the objective function is the sum of piecewise continuous functions of a single variable. This paper describes a branch-and-bound algorithm that will find a global solution to this type of problem. The algorithm has the feature that a good feasible solution is generated at the start. Moreover, at each step of the algorithm an additional feasible solution may be generated for comparison with the best solution found previously. [ABSTRACT FROM AUTHOR]

Conditions which are necessary and sufficient for convergence of a nonlinear programming algorithm are stated. It is also shown that the convergence conditions can be easily applied to most programming algorithms. As examples, algorithms by Arrow, Hurwicz and Uzawa; Cauchy; Frank and Wolfe; and Newton-Raphson are proven to converge by direct application of the convergence conditions. Also the Topkis-Veinott convergence conditions for feasible direction algorithms are shown to be a special case of the conditions stated in this paper. [ABSTRACT FROM AUTHOR]

In this paper we present an algorithm for solving mathematical programming problems of the form: Find x = (x[sub 1], ..., x[sub n]) to minimize Σφ[sub i] (x[sub i]) subject to x ε G and I ≤ x ≤ L. Each φ[sub i] is assumed to be lower semicontinuous, possibly nonconvex, and G is assumed to be closed. The algorithm is of the branch and bound type and solves a sequence of problems in each of which the objective function is convex. These problems correspond to successive partitions of the feasible set. Two different rules for refining the partitions are considered; these load to convergence of the algorithm under different requirements on the problem functions. Examples are given, and computational considerations are discussed. [ABSTRACT FROM AUTHOR]

This paper discusses the problem of assigning locations to indivisible facilities and its relation to the quadratic assignment problem. It also develops two suboptimal algorithms, one dealing with the general quadratic assignment problem and the other with an interesting special case. Qualitative and quantitative comparisons with existing suboptimal algorithms are presented. [ABSTRACT FROM AUTHOR]

In formulating linear programming problems, analysts tend to include constraints that are not binding at the optimal solution for fear of excluding necessary constraints. The inclusion of such constraints does not alter the optimum solutions, but may require many additional iterations to be taken, as well as increase the computational difficulties encountered. Most of the methods proposed to date for identification of redundant and non-binding constraints are not warranted in practice because of the excessive computations required to implement them. These methods are reviewed in the present paper, and some of them are extended. A number of additional methods are also considered. Two of these new methods are not only practical, but have proven to be powerful in solving a number of problems. These methods may be incorporated in a variant of the simplex method. After each simplex iteration is made, constraints and variables are tested and, when identified as non-binding, are eliminated. The procedure is continued, with the problem size dwindling as the algorithm progresses, until the optimum is reached. Then the values of the eliminated variables are computed by substitution into the eliminated constraints. Early evidence shows that a size reduction for small problems (having 25-30 constraints) of 50 percent is not unusual. It is hoped that the results on larger problems will be even more significant.(n4) [ABSTRACT FROM AUTHOR]

This paper considers the problem of scheduling m independent, immediately available tasks on n parallel processors. Each task has a waiting cost rate, that is a function of time, and a service time. There are no feasibility restrictions on the order in which the tasks are to be processed. An optimal scheduling rule is presented for the single processor scheduling of tasks with continuously discounted linear waiting costs. A dynamic programming algorithm has been developed for a wide class of parallel-processor problems. Linear waiting costs (with or without discounting), arbitrary processor use costs, and certain other costs are considered. In certain cases, the service time of a task may be a function of the processor that performs it. [ABSTRACT FROM AUTHOR]

The problem of scheduling n jobs on m identical processors has been introduced by R. McNaughton, but as yet no efficient algorithm has been found for determining an optimal sequencing of jobs. In this paper lower and upper bounds are given for the cost of an optimal schedule. Since the two bounds are not far apart, they may be helpful in practical scheduling problems. A procedure is described for obtaining a schedule which costs less than the given upper bound. [ABSTRACT FROM AUTHOR]

This paper presents an algorithm for determining the minimum cost schedule of tasks in a critical path network in which task cost-time curves may be concave. A computer program for the case of cost-time curves that are piecewise linear in two segments is described, and a numerical example is presented. [ABSTRACT FROM AUTHOR]

In this paper a search scheme for finding the maximum of a discrete objective function of several variables is proposed. The scheme is sequential and optimal in the sense that it determines the maximum with minimum number of measurements of the function. Unlike other sequential search techniques with this property the scheme places no condition on the number of possible maxima of the objective function. Instead it requires that some bounds on the rate of change of the function per each variable be known. The points at which measurements of the value of the objective function are to be taken are determined successively by maximization of an auxiliary function updated after each measurement. The algorithm is described in detail, and illustrated by two simple examples. [ABSTRACT FROM AUTHOR]

This paper discusses a highly effective heuristic procedure for generating optimum and near-optimum solutions for the symmetric travelling-salesman problem. The procedure is based on a general approach to heuristics that is believed to have wide applicability in combinatorial optimization problems. The procedure produces optimum solutions for all problems tested, 'classical' problems appearing in the literature, as well as randomly generated test problems, up to 110 cities. Run times grow approximately as n² in absolute terms, a typical 100-city problem requires less than 25 seconds for one case (GE63S), end about three minutes to obtain the optimum with above 95 per cent confidence. [ABSTRACT FROM AUTHOR]

MILLER and WAGNER [Opns. Res. 13, 930–945 (1965)] define joint chance-constrained programming by specifying a set of consists that are joint probability measures of the extent to which constraint violations are permitted. For the special case of a random right-hand-side vector whose elements are independent random variables, they show that an equivalent deterministic concave program exists. The purpose of this paper is to generalize this result to a class of nonlinear chance-constrained programming models with joint constraints. [ABSTRACT FROM AUTHOR]

This paper presents a realistic chemical-process optimization problem in detail, and notes the fact of its equivalence to the general nonlinear programming problem. The eight equality constraints that arise are handled by algebraic substitution, and the resulting four-variable problem is solved by a modified form of the Davidon-Fletcher-Powell algorithm. Details of the method are presented (including a FORTRAN computer program), and results of two typical cases are summarized. [ABSTRACT FROM AUTHOR]