Showing papers in "Operations Research in 1973"
TL;DR: This paper discusses a highly effective heuristic procedure for generating optimum and near-optimum solutions for the symmetric traveling-salesman problem based on a general approach to heuristics that is believed to have wide applicability in combinatorial optimization problems.
Abstract: This paper discusses a highly effective heuristic procedure for generating optimum and near-optimum solutions for the symmetric traveling-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 GE635, and about three minutes to obtain the optimum with above 95 per cent confidence.
3,761 citations
TL;DR: This note formulates a convex mathematical programming problem in which the usual definition of the feasible region is replaced by a significantly different strategy via set containment.
Abstract: This note formulates a convex mathematical programming problem in which the usual definition of the feasible region is replaced by a significantly different strategy. Instead of specifying the feasible region by a set of convex inequalities, fi(x) ≦ bi, i = 1, 2, …, m, the feasible region is defined via set containment. Here n convex activity sets {Kj, j = 1, 2, …, n} and a convex resource set K are specified and the feasible region is given by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$X =\{x \in R^{n}\mid x_{1}K_{1} + x_{2}K_{2} + \cdots + x_{n}K_{n} \subseteq K, x_{j}\geq 0\}$$ \end{document} where the binary operation + refers to addition of sets. The problem is then to find x ∈ X that maximizes the linear function c · x. When the res...
1,813 citations
TL;DR: In this article, the optimal control problem for a class of mathematical models in which the system to be controlled is characterized by a finite-state discrete-time Markov process is formulated.
Abstract: This paper formulates the optimal control problem for a class of mathematical models in which the system to be controlled is characterized by a finite-state discrete-time Markov process. The states of this internal process are not directly observable by the controller; rather, he has available a set of observable outputs that are only probabilistically related to the internal state of the system. The formulation is illustrated by a simple machine-maintenance example, and other specific application areas are also discussed. The paper demonstrates that, if there are only a finite number of control intervals remaining, then the optimal payoff function is a piecewise-linear, convex function of the current state probabilities of the internal Markov process. In addition, an algorithm for utilizing this property to calculate the optimal control policy and payoff function for any finite horizon is outlined. These results are illustrated by a numerical example for the machine-maintenance problem.
1,467 citations
TL;DR: The paper presents theory dealing primarily with properties of the relevant functions that result in convex programming problems, and discusses interpretations of this theory.
Abstract: This paper considers a class of optimization problems characterized by constraints that themselves contain optimization problems. The problems in the constraints can be linear programs, nonlinear programs, or two-sided optimization problems, including certain types of games. The paper presents theory dealing primarily with properties of the relevant functions that result in convex programming problems, and discusses interpretations of this theory. It gives an application with linear programs in the constraints, and discusses computational methods for solving the problems.
477 citations
TL;DR: An algorithm for solving resource-constrained network scheduling problems, a general class of problems that includes the classical job-shop-scheduling problem, that uses Lagrange multipliers to dualize the resource constraints, forming a Lagrangian problem.
Abstract: 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.
257 citations
TL;DR: In this article, the authors consider Markovian decision processes in which the transition probabilities corresponding to alternative decisions are not known with certainty, and they consider both a game-theoretic and a Bayesian formulation.
Abstract: 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 axe 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-min optimal policies. Finally, the paper discusses asymptotically Bayes-optimal policies.
192 citations
TL;DR: This paper presents a staff planning and scheduling model that has specific application in the nurse-staffing process in acute hospitals, and more general application in many other service organizations in which demand and production characteristics are similar.
Abstract: This paper presents a staff planning and scheduling model that has specific application in the nurse-staffing process in acute hospitals, and more general application in many other service organizations in which demand and production characteristics are similar. The aggregate planning models that have been developed for goods-producing organizations are not appropriate for these types of service organizations. In this paper the process for staffing services is divided into three decision levels: a policy decisions, including the operating procedures for service centers and for the staff-control process itself; b staff planning, including hiring, discharge, training, and reallocation decisions; and c short-term scheduling of available staff within the constraints determined by the two previous levels. These three planning "levels" are used as decomposition stages in developing a general staffing model. The paper formulates the planning and scheduling stages as a stochastic programming problem, suggests an iterative solution procedure using random loss functions, and develops a noniterative solution procedure for a chance-constrained formulation that considers alternative operating procedures and service criteria, and permits including statistically dependent demands. The discussion includes an example application of the model and illustrations of its potential uses in the nurse-staffing process.
184 citations
TL;DR: Conditions that guarantee an optimal location of a facility to lie in the convex hull of source and destination points are developed, based on a generalization of Kuhn's characterization of a conveX hull by dominance.
Abstract: This paper explores the nature of optimal solutions to a plant-location problem on a plane under general distance measures. It develops conditions that guarantee an optimal location of a facility to lie in the convex hull of source and destination points. The effect of restricting the solution to some predetermined set is explored. The development is based on a generalization of Kuhn's characterization of a convex hull by dominance. When a "Manhattan" norm is employed, it is shown to be sufficient to consider, as optimal locations, the finite number of "intersection points" in the convex hull.
181 citations
TL;DR: This paper presents a case study on the use of mathematical-computer models in developing operating policies for a university-health-service outpatient clinic, and compares the subsequent real-world results with those predicted by the models.
Abstract: This paper presents a case study on the use of mathematical-computer models in developing operating policies for a university-health-service outpatient clinic. Based on results predicted by the models, actual policy changes were made in the system; the paper compares the subsequent real-world results with those predicted by the models. The comparison demonstrated the validity of the models, and significant improvements were realized in the changed system. An analysis of daily arrival patterns was used to schedule more appointment patients during periods of low walk-in demand in order to smooth the overall daily arrivals. A Monte Carlo simulation model showed the effects of alternative decision rules for scheduling appointment periods during the day to increase patient throughput and physician utilization.
175 citations
TL;DR: The assignment and sequencing of single-operation jobs in a parallel-machine environment can be formulated as an assignment problem and means for reducing the size and computational difficulty of this problem are identified.
Abstract: This note shows that the assignment and sequencing of single-operation jobs in a parallel-machine environment can be formulated as an assignment problem. Means for reducing the size and computational difficulty of this problem are identified.
168 citations
TL;DR: The socio-technical revolution the authors have entered may well come to be known as the Resurrection.
Abstract: I believe we are leaving one cultural and technological age and entering another; that we are in the early stages of a change in our conception of the world, a change in our way of thinking about it, and a change in the technology with which we try to make it serve our purposes. These changes, I believe, are as fundamental and pervasive as were those associated with the Renaissance, the Age of the Machine that it introduced, and the Industrial Revolution that was its principal product. The socio-technical revolution we have entered may well come to be known as the Resurrection.
TL;DR: The paper modifies Ford and Fulkerson's maximal dynamic flow algorithm to construct a maximal dynamic network flow with a latest departure schedule and an earliest arrival schedule.
Abstract: This paper proves two properties of maximal network flows: 1 If there exist a maximal network flow with a given departure pattern at the sources and a maximal flow with a given arrival pattern at the sinks, then there exists a flow that has both this departure pattern at the sources and this arrival pattern at the sinks. 2 There exists a maximal dynamic network flow that simultaneously has a latest earliest departure schedule at the sources and an earliest latest arrival schedule at the sinks. The paper modifies Ford and Fulkerson's maximal dynamic flow algorithm to construct a maximal dynamic network flow with a latest departure schedule and an earliest arrival schedule.
TL;DR: In this article, an optimal control problem for the dynamics of the Vidale-Wolfe advertising model is considered, the optimal control being the rate of advertising expenditure to achieve a terminal market share within specified limits in a way that maximizes the present value of net profit streams over a finite horizon.
Abstract: This paper considers an optimal-control problem for the dynamics of the Vidale-Wolfe advertising model, the optimal control being the rate of advertising expenditure to achieve a terminal market share within specified limits in a way that maximizes the present value of net profit streams over a finite horizon. First, the special polar cases of fixed and free end points are solved with and without an upper limit on advertising rate. The complete solution to the general problem is then constructed from these polar cases. The fixed-end-point case with no upper limit on the advertising rate is solved by using Green's theorem, while the other cases require additional use of switching-point analysis based on the maximum principle. The optimal control is characterized by a combination of bang-bang, impulse, and singular control, with the singular arc forming a turnpike.
TL;DR: This study generalizes the formulation of symmetric duality to include the case where the constraints of the inequality type are defined via closed convex cones and their polars, and shows that every strongly convex function achieves a minimum value over anyclosed convex cone at a unique point.
Abstract: In this study we generalize the formulation of symmetric duality introduced by Dantzig, Eisenberg, and Cottle to include the case where the constraints of the inequality type are defined via closed convex cones and their polars. The new formulation retains the symmetric properties of the original programs. Under suitable convexity/concavity assumptions we generalize the known results about symmetric duality. The case where the function involved is strongly convex/strongly concave is also treated and Karamardian's result in this case is generalized. As a result, we show that every strongly convex function achieves a minimum value over any closed convex cone at a unique point. Some special cases of symmetric programs are then considered, leading to generalizations of Wolfe's duality as well as generalizations of quadratic and linear programming formulations.
TL;DR: This paper presents an algorithm, based on the simplex routine, that provides a way to solve a problem in which the objective function is not linear, but rather is represented by a ratio of two linear functions.
Abstract: This paper presents an algorithm, based on the simplex routine, that provides a way to solve a problem in which the objective function is not linear, but rather is represented by a ratio of two linear functions. This algorithm has a computational advantage over two previous ones because it requires neither variable transformations nor the introduction of new variables and constraints.
TL;DR: Rules are given that enable the transformation of a0-1 polynomial programming problem into a 0-1 linear programming problem to be effected with reduced numbers of constraints.
Abstract: This paper gives rules that enable the transformation of a 0-1 polynomial programming problem into a 0-1 linear programming problem to be effected with reduced numbers of constraints. Rules are also given that provide reduced numbers of variables when the true variables of interest are not individual cross-product terms, but sums of such terms or polynomials of the form âxjp.
TL;DR: A new method of generating all vertices of a given convex polytope is described, which embeds the givenpolytope in a one-higher-dimensional space and associates a number with each interior point that facilitates the construction of a spanning tree for all of the interior points.
Abstract: This paper describes a new method of generating all vertices of a given convex polytope. Additionally, irrelevant constraints are easily identified without the necessity of enumerating any of the vertices of the given convex polytope. The method embeds the given polytope in a one-higher-dimensional space. The projection of the additional vertices formed by the embedding process into the original space lie in the interior of the polytope and have a tree structure for one and two polytopes. For higher dimensions, the embedding process associates a number with each interior point that facilitates the construction of a spanning tree for all of the interior points. The interior points added can be efficiently generated by a variant of the simplex method. The vertices of the original polytope can be generated easily from these internal points by analyzing the appropriate simplex tableaux.
TL;DR: It is shown that no computing device can be programmed to compute the optimum criterion value for all problems in this class of integer programming problems in which squares of variables may occur in the constraints.
Abstract: This paper studies a class of integer programming problems in which squares of variables may occur in the constraints, and shows that no computing device can be programmed to compute the optimum criterion value for all problems in this class.
TL;DR: In this paper, a closed queuing network with a fixed number of jobs is considered and an approximate model for the open system where the computer is fed by randomly arriving jobs is developed.
Abstract: In a multiprogramming computer system several jobs may be handled simultaneously by utilizing the central processing unit in processing one job while other jobs are being served by the peripheral devices. Our analysis of such a computer system proceeds in the following way. First we view the system as a closed queuing network with a fixed number of jobs, and obtain exact results for customer cycle times and server utilization. Then we use these results to develop an approximate model for the open system where the computer is fed by randomly arriving jobs. For the approximate model we obtain simple closed-form expressions for the delay distribution. The approximate model is then tested by comparing its result for the expected sojourn time to the exact results that we obtain for some special cases.
TL;DR: The characterization of directional derivatives for three major types of extremal-value functions is reviewed and the characterization for the completely convex case is used to construct a robust and convergent feasible direction algorithm.
Abstract: Several techniques in mathematical programming involve the constrained optimization of an extremal-value function. Such functions are defined as the extremal value of a related parameterized optimization problem. This paper reviews and extends the characterization of directional derivatives for three major types of extremal-value functions. The characterization for the completely convex case is then used to construct a robust and convergent feasible direction algorithm. Such an algorithm has applications to the optimization of large-scale nonlinear decomposable systems.
TL;DR: The paper concludes that the hypothetical individual investor is representative of a large class of investors and a new derivation of the well known random-walk theory of stock-price movements leads to an improved understanding of the model parameters by relating the variance of the random- walk process to the risk aversion of the investors.
Abstract: This paper uses the principle of maximum entropy to construct a probability distribution of future stock price for a hypothetical investor having specified expectations. The result obtained is in good agreement with observations recorded in the literature. Thus, the paper concludes that the hypothetical individual investor is representative of a large class of investors. This new derivation of the well known random-walk theory of stock-price movements leads to an improved understanding of the model parameters by relating the variance of the random-walk process to the risk aversion of the investors. A practical use of the model is proposed to help the investor form an objective opinion of his skill.
TL;DR: This paper investigates the use of a discrete-time semi- Markov process to model a system that deteriorates in usage and generates semi-Markov decision processes so that optimal policies can be obtained by the policy-iteration method.
Abstract: This paper investigates the use of a discrete-time semi-Markov process to model a system that deteriorates in usage. Replacement rules that are 1 state-dependent, 2 state-age-dependent, and 3 age-dependent are proposed. The system operating costs and replacement costs are functions of the underlying states. The optimization criterion is the expected average cost per unit time. Under the first two replacement rules, the paper generates semi-Markov decision processes so that optimal policies can be obtained by the policy-iteration method. Sufficient conditions for the existence of an optimal control-limit state-dependent replacement rule are derived. For the age-dependent policy, the objective function is obtained so that the minimization can be carried out over the integers. An illustrative example is given at the end.
TL;DR: The utility of the framework is indicated and hence the importance of the original Young and Balas ideas by specifying a variety of new cuts that can be obtained from it.
Abstract: This note focuses on two new and related cut strategies for integer programming: the "convexity-cut" and "cut-search" strategies. The fundamental notions underlying the convexity-cut approach are due to Richard D. Young and Egon Balas, whose "hypercylindrical" and "intersection" cuts provide the conceptual starting points for the slightly more general framework developed here. We indicate the utility of our framework and hence the importance of the original Young and Balas ideas by specifying a variety of new cuts that can be obtained from it. The second new strategy, cut search, shares with the convexity-cut strategy the notion of generating a cut by passing a hyper-plane through the terminal endpoints of edges extended from the vertex of a cone. However, whereas the convexity-cut approach determines the extensions of these edges by reference to a convex set that contains the vertex of the cone in its interior, the cut-search approach determines these extensions by reference to associations between certain "proxy" sets of points e.g., collections of hyperplanes and "candidate solutions" to the integer program. The cut-search approach typically involves more work than the convexity-cut approach, but offers the chance to identify feasible solutions in the process, and can sometimes also yield somewhat stronger cuts than the convexity cuts.
TL;DR: 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 algorithmfor the global minimizations of a concave function subject tolinear inequality constraints.
Abstract: 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.
TL;DR: In this paper, a dynamic programming formulation for finding the sequencing of a finite set of expansion projects that meets a deterministic demand projection at minimum discounted cost is presented. And conditions for decomposition or direct solution of the sequencing problem are derived.
Abstract: This paper examines the problem of finding the sequencing of a finite set of expansion projects that meets a deterministic demand projection at minimum discounted cost. For the particular situation defined, the tuning for a next expansion may be determined directly from knowledge of projected demand and present capacity. A dynamic programming formulation for sequencing projects is developed, and solution of an example problem demonstrates that other methods proposed for this problem are not, in general, valid. Extensions of the dynamic programming formulation to include interdependence between projects and joint selection of project scale and sequencing are indicated, and conditions for decomposition or direct solution of the sequencing problem are derived.
TL;DR: This paper considers an n-phase generalization of the typical M/M/1 queuing model, where the queuing-type birth-and-death process is defined on a continuous-time n-state Marker chain, and observes that closed-form results for the limiting probabilities are difficult to obtain, if at all possible.
Abstract: This paper considers an n-phase generalization of the typical M/M/1 queuing model, where the queuing-type birth-and-death process is defined on a continuous-time n-state Marker chain. It shows that many models analyzed in the literature can be considered special cases of this framework. The paper focuses on the steady-state regime, and observes that, in general, closed-form results for the limiting probabilities are difficult to obtain, if at all possible. Hence, numerical methods should be employed. For an interesting special case, explicit results are obtained that are analogous to the classical solutions for the simple M/M/1 queue.
TL;DR: It is shown that an optimum solution to this special transportation problem is a basic feasible solution to a slightly different standard transportation problem.
Abstract: This paper considers a special class of transportation problems in which the needs of each user are to be supplied entirely by one of the available sources. We first show that an optimum solution to this special transportation problem is a basic feasible solution to a slightly different standard transportation problem. A branch-and-bound solution procedure for finding the desired solution to the latter is then presented and illustrated with an example. We then consider an extension of this problem by allowing the possibility of increasing at a cost the source capacities. The problem formulation is shown to provide a generalization to the well known assignment problem. The solution procedure appears to be relatively more efficient when the number of uses greatly exceeds the number of sources.
TL;DR: This paper characterizes the system by the probabilities of its being in the up or the down state, sets up integral equations for these probabilities by identifying suitable regenerative stochastic processes, and employs the Laplace-transform technique to solve these equations.
Abstract: This paper deals with the availability and the reliability of a two-unit system with a warm standby and subject to a single repair facility. It assumes the failure times of the units to be exponentially distributed with parameters λ and λ1, respectively. Initially, a unit is switched on and the other one is kept as a warm standby. The system breaks down if a unit fails while the other is still under repair. This paper characterizes the system by the probabilities of its being in the up or the down state, sets up integral equations for these probabilities by identifying suitable regenerative stochastic processes, and employs the Laplace-transform technique to solve these equations. It obtains the mean down-time of the system during 0, t and the mean time to system failure explicitly.
TL;DR: This paper derives the distribution of the number of busy servers seen by the two streams, and outlines a numerical procedure for computing the distributions.
Abstract: When two independent streams of customers compete for the same servers, as happens for example when first-routed and overflow traffic streams share a single trunk-group, the state of the system seen by the two different types of arriving customers will in general be different. In a loss system with mixed renewal and Poisson inputs, the blocking experienced by the renewal-type customers can be markedly different from that seen by the Poissonian customers. This paper derives the distribution of the number of busy servers seen by the two streams, and outlines a numerical procedure for computing the distributions. Examples of blocking probabilities for two cases of the renewal stream are given.
TL;DR: This paper shows that the elimination method of Szwarc removes at least as many solutions as any other method, and is therefore optimal, and how to construct a general counterexample to any procedure that removes more sequences than this optimal method.
Abstract: For solving the flow-shop scheduling problem, this paper examines elimination techniques that reduce the set of solutions to a subset that must contain the optimal solution being sought. The paper shows 1 that the elimination method of Szwarc [Naval Res. Log. Quart. 18, 295-305 1971] removes at least as many solutions as any other method, and is therefore optimal, and 2 how to construct a general counterexample to any procedure that removes more sequences than this optimal method.