Showing papers in "Mathematics of Operations Research in 1982"
TL;DR: The main focus of this paper is on determining the extent to which matching procedures can be designed which give agents the incentive to honestly reveal their preferences, and which produce stable matches.
Abstract: This paper considers some game-theoretic aspects of matching problems and procedures, of the sort which involve matching the members of one group of agents with one or more members of a second, disjoint group of agents, ail of whom have preferences over the possible resulting matches. The main focus of this paper is on determining the extent to which matching procedures can be designed which give agents the incentive to honestly reveal their preferences, and which produce stable matches. Two principal results are demonstrated. The first is that no matching procedure exists which always yields a stable outcome and gives players the incentive to reveal their true preferences, even though procedures exist which accomplish either of these goals separately. The second result is that matching procedures do exist, however, which always yield a stable outcome and which always give all the agents in one of the two disjoint sets of agents the incentive to reveal their true preferences.
908 citations
TL;DR: It is shown that the determination of a minimum cut-set of odd cardinality in a graph with even and odd vertices can be dealt with by a minor modification of the polynomially bounded algorithm of Gomory and Hu for multi-terminal networks.
Abstract: We show that the determination of a minimum cut-set of odd cardinality in a graph with even and odd vertices can be dealt with by a minor modification of the polynomially bounded algorithm of Gomory and Hu for multi-terminal networks. We connect this problem to the problem of identifying a matching or blossom constraint that chops off a point which is not contained in the convex hull of matchings or proving that no such inequality exists. Both the b-matching problems without and with upper bounds are considered. We discuss how the results of this paper can be used in conjunction with commercial LP packages lo solve b-matching problems.
406 citations
TL;DR: The generalized quasi-variational inequality problem is introduced and a theory for the existence of solution is developed and relationship with a certain implicit complementarity problem is studied.
Abstract: In this paper, we introduce the generalized quasi-variational inequality problem and develop a theory for the existence of solution. This new problem includes as special cases two existing generalizations of the classical variational inequality problem. Relationship with a certain implicit complementarity problem is also studied.
258 citations
TL;DR: A class of characteristic function games arising from maximum flow problems is introduced and is shown to coincide with the class of totally balanced games.
Abstract: A class of characteristic function games arising from maximum flow problems is introduced and is shown to coincide with the class of totally balanced games. The proof relies on the max flow-min cut theorem of Ford and Fulkerson and on the observation that the class of totally balanced games is the span of the additive games with the minimum operation.
233 citations
TL;DR: A worst-case analysis for two greedy heuristics for the integer programming problem minimize cx, Ax (ge) b, 0 (le) x (le), u, x integer, where the entries in A, b, and c are all nonnegative.
Abstract: We give a worst-case analysis for two greedy heuristics for the integer programming problem minimize cx, Ax ≥ b, 0 ≤ x ≤ u, x integer, where the entries in A, b, and c are all nonnegative. The first heuristic is for the case where the entries in A and b are integral, the second only assumes the rows are scaled so that the smallest nonzero entry is at least 1. In both cases we compare the ratio of the value of the greedy solution to that of the integer optimal. The error bound grows logarithmically in the maximum column sum of A for both heuristics.
202 citations
TL;DR: In this note, cost allocation procedures are considered and it is shown that there is exactly one such procedure which possesses four desirable properties.
Abstract: Often when several goods or services are produced by a common process or organization some estimate of the per-unit costs of these goods is required. In this note we consider cost allocation procedures and show that there is exactly one such procedure which possesses four desirable properties. Both the methods and the result were inspired by the recent work of Aumann and Shapley on nonatomic games.
194 citations
TL;DR: A slightly more specialised model is examined that generalises the location problems of interest, but now also includes the continuous aspects of the problems missing from the earlier model, namely the problem: max{wy: Σj=1najyj ≤ b, 0 ≤ yj < 1} of maximising a real-valued nondecreasing submodular function subject to a knapsack constraint.
Abstract: Recently it has been shown how various results on the behaviour of exact and approximate primal algorithms for certain discrete location problems can be obtained by studying the more general problem of maximising a submodular set function Unfortunately, this more general model tells us little about an important aspect of the discrete location problems, their linear programming relaxations and the possible use of dual solutions in obtaining bounds
Here we examine a slightly more specialised model that again generalises the location problems of interest, but now also includes the continuous aspects of the problems missing from the earlier model, namely the problem: max{wy: Σj=1najyj ≤ b, 0 ≤ yj < 1} of maximising a real-valued nondecreasing piecewise linear, concave submodular function subject to a knapsack constraint We show that a continuous greedy heuristic always attains at least 1-e-1 × 100% of the optimal value, and that for the discrete problem an adapted greedy heuristic always attains 35% of the optimal value
Specialised to the capacitated and uncapacitated location problems these results permit us to strengthen earlier results, and to obtain new results on dual greedy heuristics
154 citations
TL;DR: It is shown that the proposed price mechanism can be justified on economic terms since it is uniquely determined by a set of axioms involving only cost functions and quantities consumed and not any notion of game theory.
Abstract: We propose here a new approach for equitable cost sharing pricing based upon the Shapley value for nonatomic games. However, it is shown that the proposed price mechanism can be justified on economic terms since it is uniquely determined by a set of axioms involving only cost functions and quantities consumed and not any notion of game theory. Moreover, taking into account the utilities of the consumers, one can prove the existence of an equilibrium under this price mechanism for a general class of cost functions. This approach has the advantage of not involving any interpersonal comparisons of utilities.
142 citations
TL;DR: The number of augmentations required to achieve a maximal value flow is bounded by the cube of the number of arcs in the network, provided each successive augmentation is made along a shortest augmenting path, with ties between shortest paths broken by lexicography.
Abstract: In the “classical” network flow model, flows are constrained by the capacities of individual arcs. In the “polymatroidal” network flow model introduced in this paper, flows are constrained by the capacities of sets of arcs. Yet the essential features of the classical model are retained: the augmenting path theorem, the integral flow theorem and the max-flow min-cut theorem arc all shown to yield to straightforward generalization. We describe a maximal flow algorithm which finds augmenting paths by labeling arcs instead of nodes, as in the case of the classical model. As a counterpart of a known result for the classical model, we prove that the number of augmentations required to achieve a maximal value flow is bounded by the cube of the number of arcs in the network, provided each successive augmentation is made along a shortest augmenting path, with ties between shortest paths broken by lexicography.
119 citations
TL;DR: A certain variant of the Simplex Method, which is particularly suited for theoretical considerations is investigated, and upper and lower bounds for the expectation values of the number of steps required in phase II of theSimplex Method are deduced.
Abstract: This paper is concerned with the average number of pivot steps of the Simplex Method which are required to solve linear programming problems of the following kind: $$\mbox{max} v^{T}x,\quad \mbox{subject to} a^{T}_{i}x \leq 1 \quad \mbox{for } i=1,\ldots, m \mbox{ where } a_{i},v,x \in \mathbb{R}^{n}$$ We investigate a certain variant of the Simplex Method, which is particularly suited for theoretical considerations. For every pair m, nm ≥ n we introduce probability spaces whose random samples are the linear programming problems of the above mentioned type. The vectors ai, v are supposed to be distributed independently, identically and symmetrically under rotations on a bounded subset of Rn. We deduce upper and lower bounds for the expectation values of the number of steps required in phase II of the Simplex Method. The evaluation of these bounds yields the following result for m → ∞ and for fixed n:
1 The expectation values become Om1/n-1.
2 There exists a distribution whose expectation values increase like m1/n-1.
3 For any distribution which satisfies our conditions the expectation values tend to infinity.
4 For every e > 0 there are distributions whose expectation values become Ome.
112 citations
TL;DR: The Perron-Frobenius Theorem says that if A is a nonnegative square matrix some power of which is positive, then there exists an x0 such that Anx/‖Anx‖ converges to xn for all x > 0.
Abstract: The Perron-Frobenius Theorem says that if A is a nonnegative square matrix some power of which is positive, then there exists an x0 such that Anx/‖Anx‖ converges to xn for all x > 0. There are many classical proofs of this theorem, all depending on a connection between positively of a matrix and properties of its eigenvalues. A more modern proof, due to Garrett Birkhoff, is based on the observation that every linear transformation with a positive matrix may be viewed as a contraction mapping on the nonnegative orthant. This observation turns the Perron-Frobenius theorem into a special ease of the Banach contraction mapping theorem. Furthermore, it applies equally to linear transformations which are positive in a much more general sense. The metric which Birkhoff used to show that positive linear transformations correspond to contraction mappings is known as Hilbert's projective metric. The definition of this metric is rather complicated. It is therefore natural to try to define another, less complicated m...
TL;DR: An algorithm based on the principle of simplicial approximation is introduced to compute fixed points of upper semicontinuous point to set mappings from the product space S of unit simplices into itself, yielding a good approximation.
Abstract: In this paper an algorithm based on the principle of simplicial approximation is introduced to compute fixed points of upper semicontinuous point to set mappings from the product space S of unit simplices into itself. The algorithm is a modification of an algorithm, introduced in an earlier paper. The main feature is that it starts with an arbitrary chosen point in S and that the triangulation of S depends on the starting point. Moreover, the algorithm can terminate with a non-full-dimensional subsimplex, yielding a good approximation. An application is given for non cooperative n person games, where S is the strategy space. Some computational experiences are given.
TL;DR: The problem of maximizing the flow in a network with time-varying are capacities and storage at the nodes is formulated and solved in continuous time, and a continuous version of the Ford-Fulkerson theorem is proved, and an analogue of the labelling algorithm developed.
Abstract: The solution in discrete time of the problem of maximizing the flow in a network with time-varying are capacities and storage at the nodes is a straightforward extension of the static case. In this paper the problem is formulated and solved in continuous time. A continuous version of the Ford-Fulkerson theorem is proved, and an analogue of the labelling algorithm developed. An example is given to clarify some of the ideas of the paper and the duality theory for this problem is discussed.
TL;DR: It is shown that the problem of constructing optimal mean finishing time preemptive and nonpreemptive schedules is NP-hard, even when all nonzero tasks have identical execution time requirements.
Abstract: The problem of preemptively and nonpreemptively scheduling a set of n independent jobs on an m machine open shop, flow shop or job shop is studied. It is shown that the problem of constructing optimal mean finishing time preemptive and nonpreemptive schedules is NP-hard. These problems are not only NP-hard in the strong sense, but remain NP-hard even when all nonzero tasks have identical execution time requirements. These results will also apply to the case when the problem is to construct an optimal finish time preemptive and nonpreemptive schedule for a flow shop or a job shop. We also discuss the problem of constructing no-wait schedules for these problems.
TL;DR: The problem of scheduling a two-machine unit-operation-time jobshop to complete all jobs as rapidly as possible is shown to be solved by the following rule.
Abstract: The problem of scheduling a two-machine unit-operation-time jobshop to complete all jobs as rapidly as possible is shown to be solved by the following rule. Select for service from available jobs at each stage one with longest remaining processing time. The running time and storage space of the rule are both linear functions of the total number of operations, thereby establishing that the problem belongs to P.
TL;DR: The main result of this paper is the constructive establishment of the existence of a transition matrix whose repeated use will guarantee, for each coordinate, the achievement of the best geometri...
Abstract: This paper considers a (multiplicative) process called branching Markov decision chains in which the output at the end of the Nth period equals the product of N nonnegative matrices chosen at the beginning of periods 1, …, N, respectively, times a positive (fixed) terminal reward vector. It is assumed that the above transition matrices are drawn out of a finite set of matrices given in product form (i.e., the rows of the matrices can be selected independently out of finite sets of nonnegative row vectors). For each coordinate s we define the geometric and algebraic growth rates, respectively, of the sth coordinate of the stream of output. These growth rates are defined so that the magnitude of the corresponding sequence is of the order αNNk where α is the geometric growth rate and k is the algebraic growth rate. The main result of this paper is the constructive establishment of the existence of a transition matrix whose repeated use will guarantee, for each coordinate, the achievement of the best geometri...
TL;DR: It is shown that life distribution properties of Ḡ are inherited as corresponding properties of Fx for each x ∈ R+.
Abstract: Assume that a device is subject to wear. Over time the wear is assumed to be an increasing Levy process (Xt). Suppose the device has a threshold Y with right-tail probability Ḡ. Let ζ be the failure time of the device and Fx be its survival probability given that X0 = x. It is shown that life distribution properties of Ḡ are inherited as corresponding properties of Fx for each x ∈ R+. Optimal replacement policies for such devices are discussed for suitably chosen cost functions when Ḡ is absolutely continuous on R+ with a bounded failure rate.
TL;DR: It is shown that Reich's results extend to overtake-free paths yielding limitingSojourn time distributions which correspond to the sum of exponential independent stages of sojourn in individual nodes.
Abstract: We consider a queueing network with types of customers. Poisson arrivals, type dependent routing and state dependent service. A number of stochastic aspects relating to sojourn times are investigated. Included are limiting state distributions imbedded at customer move times, mean sojourn times, and sojourn time distributions along overtake-free paths. It is shown that Reich's results extend to overtake-free paths yielding limiting sojourn time distributions which correspond to the sum of exponential independent stages of sojourn in individual nodes. The analysis is bused solely on considerations of population distributions through conditioning on customers' itineraries in the network.
TL;DR: Formulae for minimum volume ellipsoids that contain one-sided or two-sided cuts of a givenEllipsoid are given that may be of use in the recent ellipSOid algorithms for convex and linear programming.
Abstract: We give formulae for minimum volume ellipsoids that contain one-sided or two-sided cuts of a given ellipsoid. These formulae may be of use in the recent ellipsoid algorithms for convex and linear programming.
TL;DR: The duality theorem in multiobjective nonlinear programming problems by means of conjugate set-valued functions which were introduced in Kawasaki, H. 1981 is given.
Abstract: We give a duality theorem in multiobjective nonlinear programming problems by means of conjugate set-valued functions which were introduced in Kawasaki, H. 1981. Conjugate relations and weak subdifferentials of relations. Math. Oper. Res.6 593--607.. The duality theorem is reflexive in the sense analogous to Ekeland, I., R. Temam. 1976. Convex Analysis and Variational Problems. North-Holland, Amsterdam..
TL;DR: The main conclusion drawn is that for many traffic streams, the invariant distribution seen by customers in that stream coincides with the invariants distribution of {Xt}t>0 provided the customer in motion is excluded.
Abstract: Let {Xt}t>0 be a Markov jump process and {Tn}n=0∞ a subset of the jump epochs of the process {Xt}t>0. The main results of this paper are computable formulae for the transient and invariant distributions of {XTn}n=0∞ and {XTnI}n=0∞. We also characterize when the invariant distributions of {Xt}t>0 and {XTn}n=0∞ are the same and show that in this case {Tn}n=0∞ is a Poisson process. The results are applied to a variety of discrete-slate queueing networks to obtain their state distribution as seen by customers in arrival streams, departure streams, and traffic streams on the arcs. The main conclusion drawn is that for many traffic streams, the invariant distribution seen by customers in that stream coincides with the invariant distribution of {Xt}t>0 provided the customer in motion is excluded.
TL;DR: The conjecture that there does not exist an easy construction rule for optimal strategies in the general case is established and necessary and sufficient conditions for the existence of optimal strategies which are ultimately periodic u.p. strategies are given.
Abstract: Suppose one object is hidden in one of n boxes according to a known probability distribution p1,..., pn. The boxes are to be searched sequentially. The probability of overlooking the object if we search box k and if the object is in box k is denoted by qk. The cost of a search of box k amounts to ck. If we decide to search box k directly after box k' we have to pay some extra cost called switch cost: Tk', k. The problem is---under some reasonable assumptions---to construct a strategy with minimal expected cost. If T ≡ 0 the problem has a well-known solution. Unfortunately there are many reasons which establish the conjecture that there does not exist an easy construction rule for optimal strategies in the general case. We give necessary and sufficient conditions for the existence of optimal strategies which are ultimately periodic u.p.. Since we state bounds on the length not only of the transient phase but also of the period of optimal u.p. strategies we may construct these optimal strategies by comparing finitely many strategies if optimal u.p. strategies exist. We prove some other results to reduce the number of strategies which we have to compare. At last we present a numerical example.
TL;DR: It is shown that a variety of sequential optimization problems can be expressed as special cases of an elementary search model.
Abstract: It is shown that a variety of sequential optimization problems can be expressed as special cases of an elementary search model.
TL;DR: The problem is reformulated as a two- person zero-sum game between the inspector and an inspectee and a pair of equilibrium strategies is sought, and an explicit solution for the linear case, fd = d is obtained and its asymptotic properties N large determined.
Abstract: On a finite closed lime interval an inspector wishes lo detect an event as soon as possible after its occurrence. A loss fd is incurred if the event remains undetected for a period d. Generally, f is assumed continuous and monotone increasing. Exactly N inspections are allowed. We seek a randomized inspection policy yielding the minimax expected loss. The problem is reformulated as a two-person zero-sum game between the inspector and an inspectee and a pair of equilibrium strategies is sought. An explicit solution for the linear case, fd = d is obtained and its asymptotic properties N large determined. A computational procedure for calculating the optimal policies in the nonlinear case is presented. Two additional related games are briefly treated. The optimal inspection policies each have the salient property of being a randomization over a one parameter family of pure strategies.
TL;DR: A general method of analyzing the behavior in heavy traffic of queues with different impartial queue disciplines is described, showing a degree of robustness to departures from the “first come first served” discipline.
Abstract: A general method of analyzing the behavior in heavy traffic of queues with different impartial queue disciplines is described. There are many possible limiting waiting time distributions though all are mixtures of negative exponentials. The exponential distribution itself, however, shows a degree of robustness to departures from the “first come first served” discipline.
TL;DR: Let M n be the set of signed measures on the line with nonempty finite supports whose first n - 1 moments vanish and which satisfy a specified positivity condition.
Abstract: Let Mn be the set of signed measures on the line with nonempty finite supports whose first n − 1 moments vanish and which satisfy a specified positivity condition. Let Mn* be the minimum-support measures in Mn. Then Mn is the convex cone spanned by Mn* if n ≤ 3 but not if n ≥ 4.
TL;DR: This paper illustrates how recent stochastic comparison and continuity results can be applied to prove limit theorems by bounding the standard FCFS system by the corresponding s-server system with a cyclic assignment discipline using a recent construction of Wolff Wolff.
Abstract: A new proof is given for the Kiefer-Wolfowitz Kiefer, J., J. Wolfowitz. 1955. On the theory of queues with many servers. Trans. Amer. Math. Soc.78 1--18. theorem: In the GI/G/s queue the sequence of workload vectors, and thus also the sequence of waiting times, has a proper limiting distribution independent of the initial conditions if and only if p < 0.1 assuming the interarrival times and service times have finite means. From stochastic monotonicily, convergence is easy to establish if the system starts off empty. The limit is shown lo be proper here by bounding the standard FCFS system by the corresponding s-server system with a cyclic assignment discipline, using a recent construction of Wolff Wolff, R. W. 1977. An upper bound for multi-channel queues. J. Appl. Probab.14 884--888.. The general initial condition case is treated by appropriately bounding the original sequence above and below by sequences of countable-state Markov chains for which zero is always a regenerative state. This paper thus illustrates how recent stochastic comparison and continuity results can be applied to prove limit theorems. The bounding systems are also of special interest in regenerative simulation because they provide a way lo increase the number of regeneration points.
TL;DR: The idea is to reduce the nonlinear programming problem to a finite family of “well-behaved” nonlinear programs by perturbing the objective function in a linear fashion and perturting the right-hand side of the constraints by adding a constant.
Abstract: The purpose of this paper is to give a geometrical answer to the question to the strong second order sufficiency conditions hold at any local minimum point for almost all nonlinear programs? Our idea is to reduce the nonlinear programming problem to a finite family of “well-behaved” nonlinear programs by perturbing the objective function in a linear fashion and perturbing the right-hand side of the constraints by adding a constant. Each of the “well-behaved” nonlinear programs will consist of minimizing a Morse function on a manifold with boundary, where the Morse function has no critical points on the boundary.
TL;DR: In this paper the problem of finding an optimal replacement policy is reduced to that of determining a specified number of points on the real time axis, and a condition for identifying categories which should never be used is provided, thus reducing the problem by eliminating them.
Abstract: A system must operate for t units of time. A certain component is essential for the operation of the system and must be replaced by a new component whenever it fails. There are n types of replacement categories available (with an infinite supply of each) differing only in price and life distribution. The main problem is to select the proper category for replacement at any time a failure occurs, so as to minimize the total expected cost of running the system. In this paper the problem is studied when category life distributions have a common matrix phase type representation. The generalized Erlang and the hyperexponential distributions, as well as coherent structures of the latter distributions, are some special cases of this representation. Our main result is that in many of these cases, the problem of finding an optimal replacement policy is reduced to that of determining a specified number (at most n − 1) of points on the real time axis. Also provided is a condition for identifying categories which shou...
TL;DR: Second-order conditions are given which are sufficient for a point to be a local minimizer for certain nonlinear programming problems defined on Banach spaces.
Abstract: Second-order conditions are given which are sufficient for a point to be a local minimizer for certain nonlinear programming problems defined on Banach spaces. The functions involved in the problems are not required to be smooth or convex; indeed, they are required only to satisfy certain conditions which are weak enough to be satisfied by all locally Lipschitzian functions. The sufficiency conditions are expressed in terms of Clarke generalized gradients as extended by Rockafellar. An account is given of the connections between these results and some earlier sufficiency theorems.