Showing papers in "Mathematics of Operations Research in 1984"

Open Queueing Networks in Heavy Traffic

Abstract: This paper presents heavy traffic limit theorems for the queue length and sojourn time processes associated with open queueing networks. These limit theorems state that properly normalized sequences of queue length and sojourn time processes converge weakly to a certain diffusion as the network traffic intensity converges to unity. The limit diffusion is reflected Brownian motion on the nonnegative orthant. This process behaves like Brownian motion on the interior of its state space, and reflects instantaneously on the boundaries. The reflection direction is a constant for each boundary hyperplane.

Lipschitz Behavior of Solutions to Convex Minimization Problems

Abstract: We derive the Lipschitz dependence of the set of solutions of a convex minimization problem and its Lagrange multipliers upon the natural parameters from an inverse function theorem for set-valued maps. This requires the use of contingent and Clarke derivatives of set-valued maps, as well as generalized second derivatives of convex functions.

Conditions for the Existence of Planning Horizons

Abstract: We consider the general class of discounted problems involving sequential decision making over an unbounded horizon. Under the assumptions of a finite set of policy alternatives at each decision time and costs that are eventually uniformly bounded by some exponential, existence of an optimal infinite horizon strategy is established. Moreover, it is shown that there is a finite horizon version whose first policy decision agrees with an infinite horizon optimal policy. Under the additional assumption that the infinite horizon optimal strategy is eventually cyclic, the stronger result follows that the infinite horizon optimal solution is unique for almost all interest rates and that therefore a planning horizon exists. Algorithmic implications are explored for a restricted subclass of these problems.

Constrained Undiscounted Stochastic Dynamic Programming

Abstract: In this paper we investigate the computation of optimal policies in constrained discrete stochastic dynamic programming with the average reward as utility function. The state-space and action-sets are assumed to be finite. Constraints which are linear functions of the state-action frequencies are allowed. In the general multichain case, an optimal policy will be a randomized nonstationary policy. An algorithm to compute such an optimal policy is presented. Furthermore, sufficient conditions for optimal policies to be stationary are derived. There are many applications for constrained undiscounted stochastic dynamic programming, e.g., in multiple objective Markovian decision models.

A Note on Approximation Schemes for Multidimensional Knapsack Problems

Abstract: Polynomial and fully polynomial approximation algorithms for single-dimensional knapsack problems have been extensively studied and a number of such algorithms constructed. This note shows that the problem of finding a fully polynomial approximation algorithm for multidimensional knapsack problems is NP-hard.

On the Core and Dual Set of Linear Programming Games

Abstract: We study the relation between the core of a given LP-game and the set of payoff vectors generated by optimal dual solutions to the corresponding linear program. It is well known that the set of dual payoffs is contained in the core, and that cores of games in which players are replicated converge to the set of dual payoffs when the number of replications tends to infinity. We give a necessary and sufficient condition for finite convergence. As corollaries we strengthen a sufficient condition due to Owen and obtain new conditions as well. We also study conditions in which the core and the set of dual payoffs coincide even without replication. We give a necessary and sufficient condition for this phenomenon and present two classes of LP-games with this property which properly subsumes all examples of this type discussed in the literature.

Necessary Conditions in Nonsmooth Optimization

Abstract: The paper contains four theorems concerning first order necessary conditions for a minimum in nonsmooth optimization problems in Banach spaces: a Lagrange multiplier rule for a mathematical programming problem in which an infinite dimensional equality constraint is included in the constraints, a general maximum principle for nonsmooth optimal control problems with state constraints, and a kind of multiplier rule for mathematical programming problems which applies when only finitely many equality constraints are present but when the Lipschitz continuity assumptions are removed. A summary of relevant background results from analysis is provided.

An Application of the Aumann-Shapley Prices for Cost Allocation in Transportation Problems

Abstract: The Aumann-Shapley A-S prices are axiomatically determined on certain classes of piecewise continuously differentiable cost functions. One of these classes consists of all cost functions derived from the transportation problems and some of their generalizations. These prices are used here to allocate costs among destinations in a way that each destination will pay its “real part” in the total transportation costs. An economic transportation model is presented in which the A-S prices are compatible with consumer demands. Finally an algorithm is provided to calculate both the optimal solution and the associated A-S prices for transportation problems.

Minimum Convex Cost Dynamic Network Flows

Abstract: This paper presents and solves in polynomial time the minimum convex cost dynamic network flow problem, an infinite horizon integer programming problem which involves network flows evolving over time. The model is a finite network in which each arc has an associated transit time for flow to pass through it. An integral amount of flow is to be sent through arcs of the network in each period over an infinite horizon so as to satisfy conservation of flow from some fixed period on. Furthermore, the net amount of flow “in transit” is assumed fixed over the infinite horizon. The objective is to minimize the average convex cost per period of sending flow. This problem is a generalization of the minimum convex cost network flow problem, the maximum throughput dynamic network flow problem, and the minimum cost-to-time ratio circuit problem. Furthermore, the model has applications in periodic production and transshipment, airplane scheduling, cyclic capacity scheduling, and cyclic staffing. To solve the dynamic network flow problem, one first obtains an optimal continuous-valued flow which repeats every period. The fractional variables of this solution may be rounded in such a way as to obtain an optimal integral flow which repeats every q periods, where q is the least common denominator of the fractional parts of the continuous flow. Furthermore, this integral flow is also an optimal continuous flow.

The Hirsch Conjecture for Dual Transportation Polyhedra

Abstract: An algorithm is given that joins any pair of extreme points of a dual transportation polyhedron by a path of at most (m − 1)(n − 1) extreme edges.

On the Imbalance of Inventories in Multi-Echelon Systems

Abstract: A central depot periodically orders or produces new stock and withdraws inventory in order to allocate it among several demand points, each experiencing random demands over a finite planning horizon. This system gives rise to a dynamic program with a state space of very large dimension. We determine a near-optimal allocation policy and construct a tractable, approximate dynamic program to help determine good orders and withdrawals. This is done by exploring the effects of a measure of the imbalance of inventories at the demand points. Numerical results are included.

A Minimax Theorem

Abstract: A new, general criterion is given for ensuring that a closed saddle function has a nonempty compact set of saddlepoints. Under this criterion it is shown also that every minimaximizing sequence clusters around some saddlepoint. A comparable theorem is given for semicontinuous quasi-saddle functions. The new criterion is applied to constrained saddlepoint problems and to the Fenchel-Rockafellar duality model for constrained convex minimization. Finally, the relationship to existing saddlepoint results is explored in detail.

Multiplicative Markov Decision Chains

Abstract: Previous treatments of multiplicative Markov decision chains eg., Bellman [Bellman, R. 1957. Dynamic Programming. Princeton University Press, Princeton, New Jersey.], Mandl [Mandl, P. 1967. An iterative method for maximizing the characteristic root of positive matrices. Rev. Roumaine Math. Pures Appl.XII 1317--1322.], and Howard and Matheson [Howard, R. A., Matheson, J. E. 1972. Risk-sensitive Markov decision processes. Management Sci.8 356--369.] restricted attention to stationary policies and assumed that all transition matrices are irreducible and aperiodic. They also used a “first term” optimality criterion, namely maximizing the spectral radius of the associated transition matrix. We give a constructive proof of the existence of optimal policies among all policies under new cumulative average optimality criteria which are more sensitive than the maximization of the spectral radius. The algorithm for finding an optimal policy, first searches for a stationary policy with a nonnilpotent transition matrix, provided such a rule exists. Otherwise, the method still finds an optimal policy; though in this case the set of optimal policies usually does not contain a stationary policy! If a stationary policy with a nonnilpotent transition matrix exists, then we develop a policy improvement algorithm which finds a stationary optimal policy.

On The Marginal Function in Nonlinear Programming

Abstract: For several types of finite or infinite dimensional optimization problems the marginal function or optimal value function is characterized by different local approximations such as generalized gradients, generalized directional derivatives, directional Hadamard or Dini derivatives. We give estimates for these terms which are determined by multipliers satisfying necessary optimality conditions. When the functions which define the optimization problem are more than once continuously differentiable, then higher order necessary conditions are employed to obtain refined estimates for the marginal function. As a by-product we give a new equivalent formulation of Clarke's multiplier rule for nonsmooth optimization problems. This shows that the set of all multipliers satisfying these necessary conditions is the union of a finite number of closed convex cones.

Discretization and Weak Convergence in Markov Decision Drift Processes

Abstract: In this paper we deal with continuous time Markov decision drift processes CTMDP, which permit both controls affecting jump rates of the process and impulsive controls causing immediate transitions. Between two successive jump epochs the state of the process evolves according to a deterministic drift function. Given a CTMDP we construct a sequence of discrete time Markov decision drift processes DTMDP with decreasing distance between the successive decision epochs. Sufficient conditions are provided under which the law of the CTMDP controlled by a fixed policy is the limit in the sense of weak convergence of probability measures of the laws of the approximating DTMDF's controlled by fixed discrete time policies. The conditions concern both the parameters of the CTMDP and the relation between the discrete time and continuous time policies. An application to a maintenance replacement model is given.

Life Lengths and Association: A Dynamic Approach

Abstract: We give two sets of conditions for the association of a system's component life lengths. The conditions are dynamic in the sense that they are based on evaluating, at any given time t, the effect a failure would have on the system's future behaviour. Of basic importance is the concept “weakened by failures” introduced in the paper. Mathematically the paper is based on martingales in the case of jump processes, or marked point processes.

A Matrix Game Solution of the Single-Controller Stochastic Game

Abstract: We consider a stochastic game on finitely many states with limiting average payoff. Further we assume that the law of motion depends on the actions of one player only, say the minimizer. We show that these games have the following properties: (a) The value of the stochastic game for each state s is the same as the value of the matrix game whose rows and columns are the pure stationary strategies of the players and whose entries are the corresponding payoffs, (b) These matrix games have a common optimal strategy for the maximizer which in turn yields his optimal stationary strategy in the stochastic game, (c) If the value of a stochastic subgame obtained by deleting a column or a row in a particular state coincides with the value of the original game at that state, then it coincides at all states, (d) If all actions of the minimizer at each state are essential for optimal play, then in the transient states (under optimal play) the minimizer has only one action. In such a case the original game can be solve...

Departures from a Queue with Many Busy Servers

Abstract: To analyze networks of queues, it is important to be able to analyze departure processes from single queues. For the M/M/s and M/G/∞ models, the stationary departure process is simple (Poisson), but in general the stationary departure process is quite complicated. As a basis for approximations, this paper shows that the stationary departure process is approximately Poisson when there are many busy slow servers in a large class of stationary G/GI/s congestion models having s servers, infinite waiting room, the first-come first-served discipline, and mutually independent and identically distributed service times that are independent of a stationary arrival process. Limit theorems are proved for the departure process in a G/GI/s system in which the number of servers and the offered load (arrival rate divided by the service rate) both increase. The asymptotic behavior of the departure process depends on the way the arrival rate changes. If the arrival rate is held fixed, so that the offered load increases by ...

Optimal Repair Allocation in a Series System

Abstract: In this paper we consider a problem on the optimal assignment over time of a single repairman to failed components in a series system. The following assumptions are made. The component failure and repair times are random variables with exponential distributions, independent of the state of other components. Component failures can occur even while the system is not functioning and it is permissible to reassign the repairman among failed components without penalty. It is shown that the policy which always assigns the repairman to the failed component with the smallest failure rate, among the failed ones, maximizes the availability of the system, irrespective of the values of the repair rates.

Single Machine Scheduling with Precedence Constraints of Dimension 2

Abstract: Consider the set of tasks that are partially ordered by precedence constraints. The tasks are to be sequenced so that a given objective function will assume its optimal value over the set of feasible solutions. A subset of tasks is called feasible, if for every task in the subset, all of its predecessors are also in the subset. We present a dynamic programming solution to the problem, when the constraining partial order has a dimension ≤2. This is done by definining a “compact” labeling scheme and an efficient enumerative procedure for all the feasible subsets. In this process a new characterization is given for 2-dimensional partial orders.

Existence of Equilibrium on a Manifold

Abstract: Existence of equilibrium of a continuous preference relation p or correspondence P on a compact topological space W can be proved either by assuming acyclicity or convexity (no point belongs to the convex hull of its preferred set). Since both properties may well be violated in both political and economic situations, this paper considers instead a “local” convexity property appropriate to a “local” preference relation or preference field. The local convexity property is equivalent to the nonexistence of “local” cycles. When the state space W is a convex set, or is a smooth manifold of a certain topological type, then the “local” convexity property is sufficient to guarantee the existence of a set of critical optima.

On Random Binary Trees

Abstract: A widely used class of binary trees is studied in order to provide information useful in evaluating algorithms based on this storage structure. A closed form counting formula for the number of binary trees with n nodes and height k is developed and restated as a recursion more useful computationally. A generating function for the number of nodes given height is developed and used to find the asymptotic distribution of binary trees. An asymptotic probability distribution for height given the number of nodes is derived based on equally likely binary trees. This is compared with a similar result for general trees. Random binary trees those resulting from a binary tree sorting algorithm applied to random strings of symbols are counted in terms of the mapping of permutations of n symbols to binary trees of height k. An explicit formula for this number is given with an equivalent recursive definition for computational use. A generating function is derived for the number of symbols given height. Lower and upper bounds on random binary tree height are developed and shown to approach one another asymptotically as a function of n, providing a limiting expression for the expected height. The random binary trees are examined further to provide expressions for the expectations of the number of vacancies at each level, the distribution of vacancies over all levels, the comparisons required for insertion of a new random symbol, the fraction of nodes occupied at a particular level, the number of leaves, the number of single vacancies at each level, and the number of twin vacancies at each level. A random process is defined for the number of symbols required to grow a tree exceeding any given height. Finally, an appendix is given with sample tabulations and figures of the distributions.

Permuting elements within columns of a matrix in order to minimize maximum row sum

Abstract: We study the problem of permuting the elements within columns of a given m × n matrix A so as to minimize its maximum row sum (sum of the elements in a row). We introduce and analyze an approximation algorithm of greedy type for this NP-complete problem. We prove that our algorithm produces matrices with maximum row sums no more than 3/2 − 1/2m times greater than those found by an optimization rule. Moreover, examples are presented which achieve this relative performance. Thus, our algorithm represents a substantial improvement in that all earlier algorithms have a worst-case performance that is asymptotically twice that of an optimization rule. We verify that our algorithm requires at most O(m2n) time, which is a modest increase over the earlier algorithms requiring Θ(mn log n) time in the worst-case.

A Unifying Note on Fatou's Lemma in Several Dimensions

Abstract: A general version of Fatou's lemma in several dimensions is presented. It subsumes the Fatou lemmas given by Schmeidler Schmeidler, D. 1970. Fatou's lemma in several dimensions. Proc. Amer. Math. Soc.24 300--306., Hildenbrand Hildenbrand, W. 1974. Core and equilibria of a large economy. Princeton University Press, Princeton., Cesari-Suryanarayana Cesari, L., M. B. Suryanarayana. 1978. An existence theorem for pareto problems. Nonlinear Anal.2 225--233., and Artstein Artstein, Z. 1979. A note on Fatou's lemma in several dimensions. J. Math. Econom.6 277--282.. Also, it is equivalent to an abstract variational existence result that extends and generalizes results by Aumann-Perles Aumann, R. J., M. Perles. 1965. A variational problem arising in economics. J. Math. Anal. Appl.11 488--503., Berliocchi-Lasry Berliocchi, H., J.-M. Lasry. 1973. Integrands normales et mesures parametrees en calcul des variations. Bull. Soc. Math. France101 129--184., Artstein Artstein, Z. 1974. On a variational problem. J. Math. Anal. Appl.45 404--415., and Balder Balder, E. J. 1979. On a useful compactification for optimal control problems. J. Math. Anal. Appl.72 391--398. in several respects.

Approximation Algorithms for the Assembly Line Crew Scheduling Problem

Abstract: Given an m × n nonnegative matrix, we study the problem of independently permuting the elements in each column so as to minimize the maximum row sum. This problem is NP-hard even when we restrict it to the m × 3 case. A special case of our problem is the multiprocessor scheduling problem without precedence constraints. In this paper we give a (2 − 1/m)-algorithm for the general m × n case. We also design a 3/2-algorithm for the m × 3 case, which applies a restricted longest-processing-time-first heuristic. Finally, using an edge matching algorithm, we produce a more elaborate algorithm that guarantees a bound of 4/3 for the m × 3 case.

Tight Bounds and Probabilistic Analysis of Two Heuristics for Parallel Processor Scheduling

Abstract: We study the partitioning problem, consisting in partitioning a sublist of n positive numbers into m disjoint sublists such that the maximum sublist is minimized. This is equivalent to minimizing the completion time of n jobs on m parallel identical processors. We establish upper bounds on the deviation from optimality of two heuristics: the well-known LPT heuristic, and the on-line RLP heuristic. These bounds serve to establish a probabilistic analysis of these heuristics; for both of them, the absolute deviation from optimality remains finite, when the size of the list of numbers becomes infinite. This is a stronger result than previous convergence theorems, and it is valid whenever the processing times are iid random variables with finite mean and arbitrary distributions.

A Note on Expected Makespans for Largest-First Sequences of Independent Tasks on Two Processors

Abstract: We consider the scheduling of sets of n simultaneously available tasks on two identical processors. The task execution times are assumed to be independent samples from the uniform distribution on [0, 1]. We analyze the expected makespan schedule-length performance of two well-known largest-task-first, nonpreemptive approximation rules. With the largest-first LF rule, tasks are assigned in nonincreasing order of execution time to the processors as they become available. With the restricted largest first RLF rule, tasks are assigned in pairs, one to a processor. The larger task of a pair is the first to be scheduled. If n is odd, the last task is simply assigned to the earner finishing processor in the schedule for the first n-1 tasks. We prove that the expected makespan for LF is bounded by $$\frac{n}4 + \frac{e}{2n+1},$$ and for RLF it is precisely $$\frac{n}4 + \frac{1}{2n+1}.$$ From these results simple bounds are derived on the ratio of the expected performance of LF and RLF to the expected performance of an optimization rule. Finally, a lower bound on expected LF makespans is shown to be $$\frac{n}4 + \frac{1}{4n+1}.$$

Stochastic Dominance and Moment Inequalities

Abstract: For any distribution function (df) F, define F1 = F and Fn+1 (x) = ∫−∞x Fn(y) dy. For two df's F and G, we obtain a relationship between the behaviour of Gn(x) − Fn(x) for large x and certain inequalities involving the moments of F and G. In particular, we generalize Fishburn's theorem, which deduces such inequalities from the condition that Gn(x) − Fn(x) ≥ 0 for all x.

Equivalence of Surface Density and Average Directional Density

Abstract: The average directional density criteria for evaluating tilings is shown to be equivalent to surface density and valid for random broken paths just as for straight paths.

Duality and an Algorithm for a Class of Continuous Transportation Problems

Abstract: In this paper we treat the problem of transferring mass at least cost from one line segment to another, when there is a continuous cost function cx, y giving the cost of transferring material from the point x on the first line segment to the point y on the second. The mass has to be arranged with uniform density on the second line segment after the transfer. This is a one-dimensional form of the well-known mass-transfer problem. It is an infinite-dimensional linear program. We discuss duality theory for this problem and give an algorithm which converges to an optimal solution.