Showing papers in "Operations Research in 1970"
TL;DR: It is shown that maxπwπ = C* precisely when a certain well-known linear program has an optimal solution in integers.
Abstract: This paper explores new approaches to the symmetric traveling-salesman problem in which 1-trees, which are a slight variant of spanning trees, play an essential role. A 1-tree is a tree together with an additional vertex connected to the tree by two edges. We observe that i a tour is precisely a 1-tree in which each vertex has degree 2, ii a minimum 1-tree is easy to compute, and iii the transformation on "intercity distances" cij â Cij + πi + πj leaves the traveling-salesman problem invariant but changes the minimum 1-tree. Using these observations, we define an infinite family of lower bounds wπ on C*, the cost of an optimum tour. We show that maxπwπ = C* precisely when a certain well-known linear program has an optimal solution in integers. We give a column-generation method and an ascent method for computing maxπwπ, and construct a branch-and-bound method in which the lower bounds wπ control the search for an optimum tour.
1,448 citations
TL;DR: The relation between health program output and modern decision theory for program planning is explored, and how analytical tools are useful for fitting the results of the study into larger conceptual frameworks are shown.
Abstract: In order to develop an operational definition of health, we found it necessary first to develop the concept of function/dysfunction as a continuum, based on one's ability to carry on the usual daily activities appropriate to social roles. Then, to those operating the health system, each member of the population can be seen as belonging to one and only one state from a class of functional states that can be defined on an ordinal scale. Next, we found it necessary to assign to each state a weight defined on a cardinal scale, the set of weights for these states being called the Health Status Index (HSI). The HSI rests on value judgments, of a societal nature, expressed by the administrators responsible for policy decisions. Prognosis is then defined as the transitional probability of a change in functional state with time. Thus, the concepts “state of health” and “severity of illness” are decomposed into the parameters function/dysfunction and prognosis. Finally, together with an operational definition of ti...
430 citations
TL;DR: Discrete programming, which includes integer programming and combinatorial optimization problems, is discussed and Fibonacci search is presented as an example of a nonfinite branch-and-bound procedure employing an optimal convergence rule.
Abstract: The branch-and-bound procedure is formulated in rather general terms and necessary conditions for the branching and bounding functions are precisely specified. Results include the standard properties for finite procedures, plus several convergence conditions for infinite procedures. Discrete programming which includes integer programming and combinatorial optimization problems, is discussed and Fibonacci search is presented as an example of a nonfinite branch-and-bound procedure employing an optimal convergence rule.
328 citations
TL;DR: An approach, similar to penalty functions, for solving arbitrary mathematical programs, that is a lesser constrained problem that, in some cases, may be solved with dynamic programming.
Abstract: This paper presents an approach, similar to penalty functions, for solving arbitrary mathematical programs. The surrogate mathematical program is a lesser constrained problem that, in some cases, may be solved with dynamic programming. The paper deals with the theoretical development of this surrogate approach.
201 citations
TL;DR: The purpose of this paper is to point out the connection of Metropolis's method with the minimization problem a Markov chain whose sample averages converge with probability one to (approximately) the minimizing point (z1, …, zn).
Abstract: This paper considers the problem of minimizing a function F(x1, …, xn) over a closed, bounded region S in n-dimensional space under the assumption that there exists a unique minimizing point (z1, …, zn)ϵS. In a previous paper I represented the coordinates of the minimizing point as the limit of a ratio of integrals. The same type of ratio appears, in a different context, in statistical mechanics where a Monte Carlo method has been developed, by Metropolis et al., for its numerical evaluation. The purpose of this paper is to point out the connection of Metropolis's method with the above type of minimisation problem. The idea of the method is to associate with the minimization problem a Markov chain whose sample averages converge with probability one to (approximately) the minimizing point (z1, …, zn). The Markov chain should be easily realizable on a computer. An estimate of the error from sampling over a finite time period is given.
195 citations
TL;DR: This article presented a survey of results in preference theory with intransitive indifference and discussed them for the areas of basic preference theory, consumer preference, additive utility, qualitative probability, expected utility, and social choice.
Abstract: This paper presents a survey of results in preference theory with intransitive indifference and discusses them for the areas of basic preference theory, consumer preference, additive utility, qualitative probability, expected utility, and social choice.
188 citations
TL;DR: In this article, a branch-and-bound method was proposed to enumerate all schedules and determine the optimum for a job shop scheduling problem with both precedence constraints and resource constraints.
Abstract: This paper considers a scheduling problem that has both precedence constraints of a general form as in PERT-CPM problems and resource constraints of the form in the general job-shop scheduling problem, and gives an efficient enumerative procedure for generating all active schedules for this problem. Based on this enumerative scheme, the paper describes a branch-and-bound method for implicitly enumerating all schedules and determining the optimum; finally, it gives computational experience for the problem where the objective is minimizing the project makespan.
153 citations
TL;DR: The algorithm developed to solve the quadratic binary programming 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: 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 σ2 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.
135 citations
TL;DR: This study presents some new results on three primal-feasible computational approaches for optimizing a system composed of interrelated subsystems, which approximate or generate the optimal response functions “as needed.”
Abstract: This study presents some new results on three primal-feasible computational approaches for optimizing a system composed of interrelated subsystems. The general structure treated is the same as the principal one of the classic paper by Dantzig and Wolfe, except that convex nonlinearities are permitted, provided that the overall criterion function and coupling constraints are separable by subsystem. Each approach decentralizes the optimization by iteratively allocating system resources to the subsystems, with each subsystem computing its own optimal utilization of the given resources at each iteration. The chief obstacle to directing the resource allocation centrally toward an overall optimum is that the optimal response of each subsystem, as a function of its allowed resources, is not available explicitly. All three procedures therefore approximate or generate the optimal response functions “as needed.”
128 citations
TL;DR: Techniques for solving the problems of selection of pipe diameters in a specified pipeline network to minimize the sum of investment and operation costs are developed.
Abstract: The exploitation of offshore natural gas reserves involves several phases, including production from reservoirs, separation of byproducts, and transportation to markets. The gas, which may originate as far as 100 miles from land, must be transported through pipelines to onshore delivery points. This paper develops techniques for solving the following problems: (1) selection of pipe diameters in a specified pipeline network to minimize the sum of investment and operation costs; (2) selection of minimum-cost network structures, given gas-field location and flow requirements; (3) optimal expansion of existing pipeline networks to include newly discovered gas fields. The techniques incorporate procedures for globally optimizing pipeline diameters for fixed tree structures and heuristic procedures for generating low-cost structures.
127 citations
TL;DR: This paper establishes necessary and sufficient conditions for constraint set stability requiring neither convex constraint functions not conveX constraint sets, and leads to a sufficiency result for the continuity of the optimal objective values as the right-hand side varies.
Abstract: This paper establishes necessary and sufficient conditions for constraint set stability requiring neither convex constraint functions not convex constraint sets. These conditions then lead to a sufficiency result for the continuity of the optimal objective values as the right-hand side varies. Applications to quasiconvex functions are presented.
TL;DR: This paper addresses the question of how much better (i.e., how much more profitable) the authors could expect their plans to be if somehow they could know at planning time what the outcomes of the uncertain events will turn out to be.
Abstract: The problem of planning under uncertainty has many aspects; in this paper we consider the aspect that has to do with evaluating the state of information. We address ourselves to the question of how much better (i.e., how much more profitable) we could expect our plans to be if somehow we could know at planning time what the outcomes of the uncertain events will turn out to be. This expected increase in profitability is the “expected value of perfect information” and represents an upper bound to the amount of money that it would be worthwhile to spend in any survey or other investigation designed to provide that information beforehand. In many cases, the amount of calculation to compute an exact value is prohibitive. However, we derive bounds (estimates) for the value. Moreover, in the case of operations planning by linear or convex programming, we show how to evaluate these bounds as part of a post-optimal analysis.
TL;DR: Empirical equations are given that characterize the minimum expected number of looks to detect the target, and the maximum probability of detecting the target within a given number of look.
Abstract: A target moves between two regions in a Markovian fashion, the parameters of which are known to the searcher. Discrete amounts of search effort (“looks”) may be allocated to one region at a time. This paper gives equations that characterize (a) the minimum expected number of looks to detect the target, and (b) the maximum probability of detecting the target within a given number of looks. These are solved completely for special cases, and numerical approximate solutions are described for general cases.
TL;DR: This paper considers optimal design models for queuing systems in which the decision variables are the number of servers and the mean rate at which each serves, and it is shown that a single-server system is optimal for a wide class of arrival processes and service-time distributions.
Abstract: This paper considers optimal design models for queuing systems in which the decision variables are the number of servers and the mean rate at which each serves, the total cost per unit time of operating the system is the sum of a service cost per unit time and a waiting cost per unit time, and the optimality criterion is long-run expected average cost per unit time. It is shown that a single-server system is optimal for a wide class of arrival processes and service-time distributions, and for a wide variety of waiting-cost functions.
TL;DR: This paper reviews the missile-allocation-problem literature and proposes that this review methodology by applied to other appropriate areas.
Abstract: This paper reviews the missile-allocation-problem literature. The problem considered is: given an existing weapon force and a set of targets, what is the optimal allocation of weapons to targets? References are organized by type, characterized by submodel, discussed, and annotated. It is proposed that this review methodology by applied to other appropriate areas.
TL;DR: In this article, the extreme-point-ranking approach of Murty is applied to a related linear program to obtain an optimum solution to the quadratic program. But it is not shown how to apply it to a linear program with linear constraints, it is only shown that an extreme point of the set of feasible solutions is an optimal solution.
Abstract: Certain types of quadratic programs with linear constraints have the property that an extreme point of the convex set of feasible solutions is an optimal solution. This paper presents a procedure for solving these problems, it involves determining a related linear program having the same constraints, the extreme-point-ranking approach of Murty is then applied to this linear program to obtain an optimum solution to the quadratic program.
TL;DR: It is hypothesized that the environment in which innovative activity is carried on is an important determinant of the mode and effectiveness of project implementations, and how goal operationally differences will influence the strategic behavior of the managers of innovative groups is considered.
Abstract: Many of the organizational problems that are inherent in the integration of innovation-producing activities come to light in the form of difficulties experienced in achieving implementation of output. Such difficulties are common not only in operations research but in most forms of innovative activity, among them R&D. Growing out of the work that has been pursued at Northwestern in research programs on both the R&D process and the management of operations research/management science (OR/MS) activities, this paper examines the systemic causes of such implementation problems. We have hypothesized that the environment in which innovative activity is carried on is an important determinant of the mode and effectiveness of project implementations. Specifically, we have considered how goal operationally differences, such as may be found between government and business organizations, will influence the strategic behavior of the managers of innovative groups. The data in this paper come primarily from our OR/MS st...
IBM1
TL;DR: A Markovian queuing system with balking and two heterogeneous servers determines the capacity of the slower server and obtains the optimal service rates that minimize the average characteristics of the heterogeneous system.
Abstract: This paper analyses a Markovian queuing system with balking and two heterogeneous servers determines the capacity of the slower server and obtains the optimal service rates that minimize the average characteristics of the heterogeneous system. The heterogeneous system is compared with a corresponding homogeneous system and a condition showing the efficiency of the heterogeneous system is obtained, this condition involving only the traffic intensity and the service rates. A cost model is discussed and various tables and graphs representing the average characteristics of both the systems are given.
TL;DR: A theoretical model using mainly the theory of absorbing Markov chains is applied to several human-blood-issuing policies to determine the effects of the issuing policies on average inventory levels, which determine blood shortage probabilities, and on the average age of blood at the time it is transfused.
Abstract: A theoretical model using mainly the theory of absorbing Markov chains is applied to several human-blood-issuing policies. The objective of the model applications is to determine the effects of the issuing policies on average inventory levels, which determine blood shortage probabilities, and on the average age of blood at the time it is transfused. Issuing policies that issue (transfuse) fresher blood with a higher probability than older blood are defined as modified fifo policies, and issuing policies that issue older blood with a higher probability than fresher blood are defined as modified fifo policies. Application of the theoretical model to the various issuing policies allows complete evaluation of the policies, and a policy choice can be made on the basis of the evaluation.
TL;DR: The purpose of this paper is to apply the enumeration philosophy to the classical knapsack problem and shows that this approach applies quite naturally to this type of integer linear program when combined with the Fourier-Motzkin elimination method for solving linear inequalities.
Abstract: Enumeration techniques have been shown to be successful for solving integer linear programming problems. The purpose of this paper is to apply the enumeration philosophy to the classical knapsack problem; it shows that this approach applies quite naturally to this type of integer linear program when combined with the Fourier-Motzkin elimination method for solving linear inequalities. Some computational results are reported.
TL;DR: An efficient, one-pass search procedure based on bisection for a multi-item, multi-echelon inventory problem is presented, and it is shown that, in general, the minimax sequential search is bisection.
Abstract: One-constraint optimization problems are approached via Lagrange multipliers. Sequential search schemes for generating suitable trial multiplier values are compared, and it is shown that, in general, the minimax sequential search is bisection. For certain applications, it pays to design search procedures that take advantage of special structure, such as recursively defined functions. An efficient, one-pass search procedure based on bisection for a multi-item, multi-echelon inventory problem is also presented.
TL;DR: This paper is based on an address to the Philadelphia Section of the Society given on June 5, 1970 and aims to provide a history of the field and some of the principles behind its development.
Abstract: The worst thing that can happen to operations research is that our conception of what it ought to be becomes equivalent to our conception of what it is. In a changing world, even equilibrium must be dynamic.
TL;DR: The algorithms presented apply when the preference functions h(x) and g(y) are convex, and continuously differentiable, k is a convex polytope, ξ has a distribution that satisfies mild convergence conditions, and the objective is to minimize the expectation of the sum of the two preference functions.
Abstract: This paper presents computational algorithms for the solution of a class of stochastic programming problems. Let x and y represent the decision and state vectors, and suppose that x must be chosen from some set K and that y is a linear function of both x and an additive random vector ξ. If y is uniquely determined once x is chosen and ξ is observed, we say that the problem has simple recourse. The algorithms presented apply, e.g., when the preference functions h(x) and g(y) are convex, and continuously differentiable, k is a convex polytope, ξ has a distribution that satisfies mild convergence conditions, and the objective is to minimize the expectation of the sum of the two preference functions. An illustrative example of an inventory problem is formulated, and the special case when g is asymmetric, quadratic, and separable is presented in detail to illustrate the calculations involved.
TL;DR: The conditions under which the expected waiting times for all classes satisfy a simple linear equality constraint that is independent of queue discipline for a large class of disciplines are generalized.
Abstract: Kleinrock first showed that, for a multiclass M/G/1 queue, the expected waiting times for all classes satisfy a simple linear equality constraint that is independent of queue discipline for a large class of disciplines. We generalize here the conditions under which this result holds and give a simpler proof.
TL;DR: This paper analyzes a single-server queuing system in which service is alternated between two queues, assuming each queue is assumed to have an independent Poisson input and an independent general service-time distribution.
Abstract: This paper analyzes a single-server queuing system in which service is alternated between two queues. Each queue is assumed to have an independent Poisson input and an independent general service-time distribution. The alternating priority rule is followed. Independent general distributions are assumed for the intervals required to switch service from one queue to the other. The following aspects of the steady-state solution are presented for each queue: average delay of an entering customer, average length, average durations of the busy period, and average rate at which service is offered to a queue. The model has been applied to a generalized data-communications system in which transmission between two data stations is possible in only one direction at a time.
TL;DR: Several theorems giving necessary and sufficient conditions for v directed graph to be a network graph are obtains, and the results are applied to solving some problems of smoothing traffic flow.
Abstract: Let G be a directed graph such that even edge e of G is associated with a positive integer, called the index of e. Then G is called a network graph if, at every vertex v of G, the sum of the indices of the edges terminating at v is equal to that of the edges incident from v. This paper obtains several theorems giving necessary and sufficient conditions for v directed graph to be a network graph, and applies the results to solving some problems of smoothing traffic flow.
TL;DR: This note investigates the solutions obtained by various techniques such as switch and check, branch and bound with and without backtracking, modified decomposition, and rounded linear programming considering both the quality of the solutions and the computational efficiency.
Abstract: This note is concerned with the solution of the flow-shop scheduling problem where all jobs have the same machine ordering. Because of the combinatorial nature of this problem, most practical situations remain unsolved. Various techniques such as switch and check, branch and bound with and without backtracking, modified decomposition, and rounded linear programming have been proposed by several investigators. However, no comparative evaluation of these procedures has been previously made. This note investigates the solutions obtained by these procedures considering both the quality of the solutions and the computational efficiency. Extensive experimentation has been conducted and significant results are reported. The effects of changes in the size of problems on the above criteria are also included.
TL;DR: Given a mathematical program, this paper constructs an alternate problem with its feasibility region a superset of the original mathematical program so that a penalty is imposed for solutions outside the original feasibility region.
Abstract: Given a mathematical program, this paper constructs an alternate problem with its feasibility region a superset of the original mathematical program. The objective function of this new problem is constructed so that a penalty is imposed for solutions outside the original feasibility region. One attempts to choose an objective function that makes the optimal solutions to the new problem the same as the optimal solutions to the original mathematical program.
TL;DR: Two algorithms for targeting strikes in a lines-of-communication (LOC) network are presented, which attempt to make such costs as large as possible over time when the effect of targeting strikes is to increase arc-cost functions and decrease arc capacities for a given period of time.
Abstract: This paper presents two algorithms for targeting strikes in a lines-of-communication (LOC) network. The LOCs are represented by a network of nodes and directed arcs. It is assumed that the user of the LOCs is attempting to achieve a circulation flow at minimum cost, a very general goal that includes, as special cases, maximizing flow between two points, meeting required flows between two points at minimum cost, and combinations of these two. The algorithms presented here attempt to make such costs as large as possible over time when the effect of targeting strikes is to increase arc-cost functions and decrease arc capacities for a given period of time. The first algorithm treats the situation where arc costs are linear functions of flow; the second treats the situation where arc costs are piecewise linear functions of flow with one break point.
TL;DR: This paper treats a discrete-time Markov decision model with an infinite planning horizon and no discounting as a "bias-optimal" policy, which satisfies a criterion that is more selective than maximizing the gain rate.
Abstract: This paper treats a discrete-time Markov decision model with an infinite planning horizon and no discounting. A "bias-optimal" policy for this decision problem satisfies a criterion that is more selective than maximizing the gain rate. The problem of computing a bias-optimal policy, also treated by Veinott in 1966, is here parsed into a sequence of three simple Markov decision problems, each of which can be solved by linear programming or policy iteration.