Showing papers in "Mathematics of Operations Research in 1977"
TL;DR: Graph-theoretic ideas are used to analyze cooperation structures in games, and fair allocation rules are proven to be unique, closely related to the Shapley value, and stable for a wide class of games.
Abstract: Graph-theoretic ideas are used to analyze cooperation structures in games. Allocation rules, selecting a payoff for every possible cooperation structure, are studied for games in characteristic function form. Fair allocation rules are defined, and these are proven to be unique, closely related to the Shapley value, and stable for a wide class of games.
1,415 citations
TL;DR: This work considers partitioning algorithms for the approximate solution of large instances of the traveling-salesman problem in the plane, in which partitioning is used in conjunction with existing heuristic algorithms.
Abstract: We consider partitioning algorithms for the approximate solution of large instances of the traveling-salesman problem in the plane. These algorithms subdivide the set of cities into small groups, construct an optimum tour through each group, and then patch the subtours together to form a tour through all the cities. If the number of cities in the problem is n, and the number of cities in each group is t, then the worst-case error is $O\sqrt{n/t}$ . If the cities are randomly distributed, then the relative error is Ot-1/2 with probability one. Hybrid schemes are suggested, in which partitioning is used in conjunction with existing heuristic algorithms. These hybrid schemes may be expected to give near-optimum solutions to problems with thousands of cities.
451 citations
TL;DR: A simple proof of finiteness is given for the simplex method under an easily described pivoting rule and a second new finite version of thesimplex method is presented.
Abstract: A simple proof of finiteness is given for the simplex method under an easily described pivoting rule. A second new finite version of the simplex method is also presented.
417 citations
TL;DR: The method is based on combining, modifying and extending the nonsmooth optimization work of Wolfe, Leraarechal, Feuer, Poljak and Merrill and can be thought of as a generalized reset conjugate gradient algorithm.
Abstract: We present an implementable algorithm for solving constrained optmization problems defined by functions that are not everywhere differenliable. The method is based on combining, modifying and extending the nonsmooth optimization work of Wolfe, Leraarechal, Feuer, Poljak and Merrill. It can be thought of as a generalized reset conjugate gradient algorithm. We also introduce the class of weakly upper semismooth functions. These functions are locally Lipschitz and have a semicontinuous relationship between their generalized gradient sets and their directional derivatives. The algorithm is shown to converge to stationary points of the optimization problem if the objective and constraint functions are weakly upper semismooth. Such points are optimal points if the problem functions are also semiconvex and a constraint qualification is satisfied. Under stronger convexity assumptions, bounds on the deviation from optimally of the algorithm iterates are given.
199 citations
TL;DR: This work considers an extension of this special case model to include the possibility of individual task deadlines, in which case the goal is to minimize maximum lateness and shows that it makes a considerable difference whether the precedence is of “in-tree” or “out- tree” form.
Abstract: A basic problem of deterministic scheduling theory is that of scheduling n equal length tasks on m identical processors subject to precedence constraints. Although the general problem of finding a schedule which minimizes makespan is NP-complete, the important special case with “treelike” precedence constraints can be solved by a well-known algorithm of T. C. Hu. We consider an extension of this special case model to include the possibility of individual task deadlines, in which case the goal is to minimize maximum lateness. Our results show that it makes a considerable difference whether the precedence is of “in-tree” or “out-tree” form. In the former case the problem can be solved in lime On log n; in the latter it is NP-complete. We also discuss applications of our results to related scheduling problems.
113 citations
TL;DR: It is shown that valid inequalities for 0--1 problems can be essentially characterized by two underlying functions, one of which is superadditive, essential to the characterization of maximal inequalities, the projection of valid inequalities and the definition of a master polytope.
Abstract: It is shown that valid inequalities for 0--1 problems can be essentially characterized by two underlying functions, one of which is superadditive. These functions are essential to the characterization of maximal inequalities, the projection of valid inequalities and the definition of a master polytope. Similar properties are shown to hold for 0--1 group problems.
101 citations
TL;DR: The axiom of Pareto optimally in Nash's definition of a solution to the bargaining problem may be replaced by anAxiom of individual rationality, without altering the result.
Abstract: The axiom of Pareto optimally in Nash's definition of a solution to the bargaining problem may be replaced by an axiom of individual rationality, without altering the result.
98 citations
TL;DR: Sharp upper and lower bounds on the Laplace-Stieltjes transform of the corresponding distribution function are derived and can prove useful in producing conservative estimates of a system's performance and in judging the information content of a partial characterization.
Abstract: Several partial characterizations of positive random variables (e.g., certain moments) are considered. For each characterization, sharp upper and lower bounds on the Laplace-Stieltjes transform of the corresponding distribution function are derived. These bounds are then shown to be applicable to several problems in queueing and traffic theory. The results can prove useful in producing conservative estimates of a system's performance, in judging the information content of a partial characterization and in providing insight into approximations.
67 citations
TL;DR: This paper considers the problem of solving a system of n nonlinear equations in n variables, when the underlying functions are continuously differentiable and their derivative satisfies a Lipschitz condition, and shows that certain realizations of these methods achieve quadratic convergence when they reach sufficiently close to the solution.
Abstract: This paper considers the problem of solving a system of n nonlinear equations in n variables, when the underlying functions are continuously differentiable and their derivative satisfies a Lipschitz condition. We restrict our attention to methods which are based on complementary pivoting, also known as fixed point algorithms. We show that certain realizations of these methods achieve quadratic convergence when they reach sufficiently close to the solution. And in these cases, without using the derivatives of the mappings, the computational work involved is comparable to that of Newton's method.
67 citations
TL;DR: This paper considers the assessment of von Neumann-Morgenstern utility functions u defined on two attributes from the viewpoint of mathematical approximation theory and focuses on approximations v of u that can be written as vx, y = f1xg1y +... + fmxgmy where the fi and/or gi are based on single-attribute conditional utility functions.
Abstract: This paper considers the assessment of von Neumann-Morgenstern utility functions u defined on two attributes from the viewpoint of mathematical approximation theory. It focuses on approximations v of u that can be written as vx, y = f1xg1y +... + fmxgmy where the fi and/or gi are based on single-attribute conditional utility functions. The paper does not rely on independence axioms that enable ux, y to be expressed in simplified forms when they are valid.
58 citations
TL;DR: A model for a general single server queueing system operating under the earliest due date scheduling rule is investigated, which can be interpreted more generally as a dynamic priority queue-discipline, which is found to be analogous to the expressions derived by Cobham in the special case of static priorities.
Abstract: We investigate a model for a general single server queueing system operating under the earliest due date scheduling rule, which can be interpreted more generally as a dynamic priority queue-discipline. We consider both preemptive resume and non preemptive disciplines within this framework. No assumptions are made about the distributions of interarrival times and service times of customers in this model, so that characterizations can be derived for arbitrary systems. The analysis depends on suitably defining the concepts of the server's workload and busy period of the system with respect to the priority classes involved. Equations are developed for the virtual waiting times of customers arriving at the system at any time t, and these imply properties of virtual lateness. In the case of an M/G/1 system, expressions for mean waiting times in transient and steady state are derived, and these are found to be analogous to the expressions derived by Cobham in the special case of static priorities. Bounds for exp...
TL;DR: It is shown that on two or more machines, the problem is NP-complete even if the precedence constraints are tree-like, and this work proves the result both for in-trees in which the root is the last task to be processed, and out-Trees inWhich theroot is the first task to being processed.
Abstract: There are n tasks lo be scheduled for processing on a set of identical parallel machines We are interested in minimizing the mean flow time, which is related to the sum of the finishing times of all tasks When tasks can be processed in any order, optimal schedules can be constructed in On log n time on any number of identical machines With arbitrary precedence constraints the problem becomes NP-complete even on one machine However, for series-parallel precedence constraints an On log n algorithm is known for one machine We show that on two or more machines, the problem is NP-complete even if the precedence constraints are tree-like We prove the result both for in-trees in which the root is the last task to be processed, and out-trees in which the root is the first task to be processed
TL;DR: A countable stage, countable state, finite action decision problem is considered where the objective is the maximization of the expectation of an arbitrary utility function defined on the sequence of states.
Abstract: A countable stage, countable state, finite action decision problem is considered where the objective is the maximization of the expectation of an arbitrary utility function defined on the sequence of states. Basic concepts are formulated, generalizing the standard notions of the optimality equations, conserving and unimprovable strategies, and strategy and value iteration. Analogues of positive, negative and convergent dynamic programming are analyzed.
TL;DR: It is obtained necessary and sufficient conditions which guarantee that the maximal total expected reward for a planning horizon of n epochs minus n times the long run average expected reward has a finite limit as n → ∞ for each initial state and each final reward vector.
Abstract: This paper considers undiscounted Markov Decision Problems. For the general multichain case, we obtain necessary and sufficient conditions which guarantee that the maximal total expected reward for a planning horizon of n epochs minus n times the long run average expected reward has a finite limit as n → ∞ for each initial state and each final reward vector. In addition, we obtain a characterization of the chain and periodicity structure of the set of one-step and J-step maximal gain policies. Finally, we discuss the asymptotic properties of the undiscounted value-iteration method.
TL;DR: The study of sequential decision problems with expected utility criteria is continued and the definition of stationarity is seen to be inadequate for generalizations of “average reward” criteria.
Abstract: The study of sequential decision problems with expected utility criteria is continued. Stationary problems are defined similarly to stationary problems with additive or separable utility, except that an additional requirement, stationarity of the preference structure, is imposed. The optimality of stationary and memoryless strategies is investigated for problems with upper and lower convergent utility or which are upper or lower transient. By example, the definition of stationarity is seen to be inadequate for generalizations of “average reward” criteria. Simplifications in the strategy iteration algorithm which result from stationarity are discussed.
TL;DR: Theorem 2 shows that if the network's period is one and if node i is contained in a cycle of length n, then the ith row and column of Pt are positive for t ≥ N-1n.
Abstract: Let P be an irreducible N × N matrix having nonnegative entries, and consider the directed network containing arc i, j if and only if Pij is positive. Theorem 1 expresses the period of this network in terms of any of its arborescences. Theorem 2 shows that if the network's period is one and if node i is contained in a cycle of length n, then the ith row and column of Pt are positive for t ≥ N-1n. Theorem 3 shows how to compute the network's period with work proportional to the number of its arcs.
TL;DR: The purpose of this paper is to determine the optimal replacement policy among the class of policies that replace at shock times, and the main assumption is that the cumulative damage process is a semi-Markov process.
Abstract: Consider a system that is subject to a sequence of randomly occurring shocks; each shock causes some damage of random magnitude to the system. Any of the shocks might cause the system to fail, and the probability of such a failure is a function of the sum of the magnitudes of damage caused by all previous shocks. When the system fails, it must be immediately replaced and a failure cost is incurred. If the system is replaced before failure, a smaller replacement cost is incurred, and that cost may depend upon the state of the system at replacement time.
The purpose of this paper is to determine the optimal replacement policy among the class of policies that replace at shock times. The main assumption is that the cumulative damage process is a semi-Markov process. The cost criterion is to minimize the discounted cost of replacement. See Feldman [Feldman, R. M. 1976. Optimal replacement with semi-Markov shock models. J. Appl. Probab.13 108--117; Feldman, R. M. 1977. The maintenance of systems governed by semi-Markov shock models. Proceedings of the Conference on the Theory and Applications of Reliability with Emphasis on Bayesian and Nonparametric Methods. Edited by C. P. Tsokos. Academic Press.] for the average cost criterion case. The general approach will be to apply optimal stopping theory to the replacement problem.
TL;DR: Results of a simulation giving i the probability that all m constraints are relevant and ii a formula for the expected number of vertices of a polytope, are presented.
Abstract: A formula for the probability that a randomly generated n-polytope defined by m half-spaces is bounded is presented. Results of a simulation giving i the probability that all m constraints are relevant and ii a formula for the expected number of vertices of a polytope, are presented.
TL;DR: Some new properties of the zero set of the dam process are obtained and several types of imbedded renewal processes may be defined and various cycles in the content process may be identified.
Abstract: The classical dam with input according to a process with stationary, independent increments and output at unit rate is considered The stochastic properties of level crossings, ie up-and downcrossings of a fixed level x > 0 are studied It is shown that successive downcrossings of any such level constitute a renewal process with as strictly positive lifetimes This property allows the content process to be described as a regenerative process In fact several types of imbedded renewal processes may be defined and various cycles in the content process may be identified The number of downcrossings of a fixed level in each such cycle as well as other functionals are studied In particular, some new properties of the zero set of the dam process are obtained This study has arisen from questions in cost optimization problems of dams and a simple example is discussed to illustrate the use of the results in this context
TL;DR: A sequence of inspections for which the total cost is minimum is found, by reducing the stochastic problem lo a free boundary problem in analysis (called a quasi variational inequality), which is then solved.
Abstract: A Brownian motion ξ(t) is developing in time with cost f(ξ(t)) per unit time. It is assumed that ξ(t) = (x(t), y(t)) where x(t) and y(t) are independent Brownian motions. The component x(t) is being continuously observed, whereas the position of y(t) can be discovered only by making observations at random times σn with incurred cost β(ξ(σn)). Thus, σn is a stopping time with respect to the σ-field generated by x(t), t ≥ 0 and the random variables y(σ1), …, y(σn−1). A set A is given, and at the time σn the following policy is executed: (a) continue with the process until the next inspection, if y(σn) ∈ A, (b) stop and shut off the process with cost γ(ξ(σn) if y(σn) ∉ A. The problem considered in this paper is that of finding an optimal sequence of inspections {σn} This is done by first transforming the stochastic problem into a free boundary problem in analysis and then studying the latter.
TL;DR: It is proved that every continuous value is diagonal, which in particular implies that every value on a closed reproducing space is diagonal.
Abstract: It is proved that every continuous value is diagonal, which in particular implies that every value on a closed reproducing space is diagonal. We deduce also that there are noncontinuous values.
TL;DR: A gradient-path oriented algorithm for unconstrained minimization, requiring first derivatives of the objective, is presented and it is shown to converge to a stationary point of a continuously differentiable Function.
Abstract: A gradient-path oriented algorithm for unconstrained minimization, requiring first derivatives of the objective, is presented. The algorithm possesses the quadratic termination property and it is shown to converge to a stationary point of a continuously differentiable Function.
TL;DR: The problem of finding a solution to nonlinear equations by way of the Homotopy Invariance Theorem is considered, using continuation methods for simplicial mappings.
Abstract: The problem of finding a solution to nonlinear equations by way of the Homotopy Invariance Theorem is considered, using continuation methods for simplicial mappings.
TL;DR: Nonconvex duality properties for multiobjective optimization problems are obtained by using a characterization of Pareto optima by means of generalized Tchebycheff norms.
Abstract: Nonconvex duality properties for multiobjective optimization problems are obtained by using a characterization of Pareto optima by means of generalized Tchebycheff norms.
Bounds for the corresponding duality gap are given, and approximate Pareto multipliers are constructed. A generalized notion of Pareto multipliers for quasi-convex multiobjective problems is introduced.
TL;DR: To prove the continuity of the average costs as function on the space of strategies some perturbation results for quasi-compact linear operators are used to prove the existence of an average optimal strategy.
Abstract: In this paper stationary Markov decision problems are considered with arbitrary state space and compact space of strategies. Conditions are given for the existence of an average optimal strategy. This is done by using the fact that a continuous function on a compact space attains its minimum. To prove the continuity of the average costs as function on the space of strategies some perturbation results for quasi-compact linear operators are used. In a first set of conditions the boundedness of the one-period cost functions and the quasi-compactness of the Markov processes are assumed. In more general conditions the boundedness of the cost functions is replaced by the boundedness, on a subset A of the state space, of the recurrence time and costs until A and the quasi-compactness of the Markov processes are replaced by the quasi-compactness of the embedded Markov processes on A.
TL;DR: It is shown that if the nodes labeled by the algorithm are scanned in a consistent manner, then this algorithm will always produce a maximal flow in a finite number of iterations.
Abstract: This note fills a gap in the theory of convergence of the Ford-Futkerson flow algorithm. We show that if the nodes labeled by the algorithm are scanned in a consistent manner, then this algorithm will always produce a maximal flow in a finite number of iterations.
TL;DR: In an expected utility analysis of a decision problem, knowledge of the utility function at a few selected points may be available and sufficient conditions for stochastic orderings are presented for the following cases of partially specified utility functions: nondecreasing functions, and nondECreasing concave functions.
Abstract: In an expected utility analysis of a decision problem, knowledge of the utility function at a few selected points may be available. When combined with general properties such as monotonicity or concavity, the limited knowledge of the utility function may be sufficient for ranking a pair of probability distributions unambiguously. Necessary and sufficient conditions for such stochastic orderings are presented for the following cases of partially specified utility functions: nondecreasing functions, and nondecreasing concave functions. Examples of these orderings are presented.
TL;DR: There are strategies available for nonnegative gambling problems which are not only (epsilon)-optimal but persist in being conditionally (Epsilon-optimal along every history.
Abstract: There are strategies available for nonnegative gambling problems which are not only ϵ-optimal but persist in being conditionally ϵ-optimal along every history.
TL;DR: In this article, the equivalence of the sensitive optimality criteria as introduced by Veinott is shown, and the Laurent expansion of the total discounted expected return for the various policies is derived.
Abstract: Discrete time Markov decision processes with a countable state space are investigated. Under a condition of Liapunov function type the Laurent expansion of the total discounted expected return for the various policies is derived. Moreover, the equivalence of the sensitive optimality criteria as introduced by Veinott is shown.
TL;DR: The algorithm of this paper corrects an erroneous procedure often “discovered” by new students, and since it builds from that, this paper may thus also serve as a useful introduction to the field for students.
Abstract: One of the newest areas of mathematical programming, the calculation of fixed points, permits solution of problems unsolvable by earlier methods. The algorithm of this paper, moreover, details two theoretical improvements over previous fixed point algorithms. First, previous algorithms utilize a homotopy structure in which an additional dimension is added to the original problem. This algorithm eliminates the “additional dimension,” thereby simplifying the analysis and easing understanding. Second, previous algorithms require the points generated to conform to a fixed predetermined grid. This algorithm, however, allows the points to be selected flexibly dependent upon the actual function values encountered. The algorithm is therefore adaptive. Hopefully, this adaptive feature will speed convergence. It also suggests a new area of research, that of determining “optimum” adaption functions. To achieve all this, the algorithm utilizes an unconventional modification of the barycentric subdivision in which the points may be selected “off center” hence the name eccentric. Finally, the algorithm corrects an erroneous procedure often “discovered” by new students, and since it builds from that, this paper may thus also serve as a useful introduction to the field for students.