John H. Vande Vate

Linear programming brings marital bliss

A stable matching is an assignment of n men to n women so that no two people prefer each other to their respective spouses. This paper describes the convex hull of the incidence vectors of stable matchings. With this description, one may solve the optimal stable marriage problem as a linear program.

Incentives in Two-Sided Matching with Random Stable Mechanisms

SummaryThis paper considers the incentives confronting agents who face the prospect of being matched by some sort of random stable mechanism, such as that discussed in Roth and Vande Vate (1990). A one period game is studied in which all stable matchings can be achieved as equilibria in a natural class of undominated strategies, and in which certain unstable matchings can also arise in this way. A multi-period extension of this game is then considered in which all subgame perfect equilibria must result in stable matchings. These results suggest avenues to explore markets in which matching is organized in a decentralized way.

Balanced loading

We develop a heuristic for a problem motivated by the loading of aircraft or trucks: pack blocks into a bin so that their center-of-gravity is as close as possible to a target point. Our heuristic either produces good solutions or else signals that none is possible. It also works when loading nonhomogeneous blocks into a bin of nonzero and possibly nonhomogeneous mass.

Impulse Control of Brownian Motion: The Constrained Average Cost Case

When a manufacturer places repeated orders with a supplier to meet changing production requirements, he faces the challenge of finding the right balance between holding costs and the operational costs involved in adjusting the shipment sizes. We consider an inventory whose content fluctuates as a Brownian motion in the absence of control. At any moment, a controller can adjust the inventory level by any positive or negative quantity, but incurs both a fixed cost and a cost proportional to the magnitude of the adjustment. The inventory level must be nonnegative at all times and continuously incurs a linear holding cost. The objective is to minimize long-run average cost. We show that control band policies are optimal for the average cost Brownian control problem and explicitly calculate the parameters of the optimal control band policy. This form of policy is described by three parameters {q,Q,S}, 0 <q ≤ Q <S. When the inventory falls to zero (rises to S), the controller expedites (curtails) shipments to return it to q (Q). Employing apparently new techniques based on methods of Lagrangian relaxation, we show that this type of policy is optimal even with constraints on the size of adjustments and on the maximum inventory level. We also extend these results to the discounted cost problem.

Global Stability of Two-Station Queueing Networks

This paper summarizes results of Dai and Vande Vate [15, 14] characterizing explicitly, in terms of the mean service times and average arrival rates, the global pathwise stability region of two-station open multiclass queueing networks with very general arrival and service processes. The conditions for pathwise global stability arise from two intuitively appealing phenomena: virtual stations and push starts. These phenomena shed light on the sources of bottlenecks in complicated queueing networks like those that arise in wafer fabrication facilities. We show that a two-station open multi-class queueing network is globally pathwise stable if and only if the corresponding fluid model is globally weakly stable. We further show that a two-station fluid model is globally (strongly) stable if and only if the average service times are in the interior of the global weak stability region. As a consequence, under stronger distributional assumptions on the arrival and service processes, the queueing network is globally stable in a stronger sense when the mean service times are in the interior of the global pathwise stability region. Namely, the underlying state process of the queueing network is positive Harris recurrent.

Question-asking strategies for Horn clause systems

An expert system applies the deduction rules in its knowledge base to a set of initial data to reach a conclusion. When the initial data are insufficient, the expert system may ask the user for additional information. This paper analyzes effectiveness and efficiency of question-asking strategies in expert systems with Horn clause knowledge bases. An effective strategy reaches a conclusion after asking as few questions as possible. An efficient strategy can be computed quickly. We prove that effective strategies are, unfortunately, not efficient. However, we present a somewhat less effective but very efficient strategy. It employs an algorithm which simultaneously performs deduction and question selection in log-linear time.

Fractional matroid matchings

Abstract This paper introduces a linear relaxation of the matroid matching problem, called the fractional matroid matching problem. When the matroid matching problem is in fact a matroid intersection problem, the fractional matroid matching polytope and the matroid matching polytope coincide. When the matroid matching problem is in fact a matching problem in a graph, the fractional matroid matching polytope and the classic fractional matching polytope of the graph coincide. Thus, the fractional matroid matching polytope may properly contain the matroid matching polytope. The fractional matroid matching polytope is a lattice polyhedron and, although its extreme points are not all integral, they are half-integral. This paper establishes strong relationships between extreme points of the fractional matroid matching polytope and those of graphic fractional matching polyhedra. Despite these strong relationships, adding the rank 1 inequalities does not define the matroid matching polytope. In fact, even the matching polytope of a graphic matroid is not generally described by adding these inequalities.

A Probabilistic Analysis of Two-Machine Flowshops

We study a two-machine flowshop in which all processing times are independently and identically distributed, with values known to the scheduler. We are able to describe in detail the expected behavior of the flowshop under optimal and heuristic schedules. Our results suggest that minimizing makespan might be a superfluous objective: random schedules are easier to construct and require significantly less intermediate storage between the machines; moreover, they are known to be asymptotically optimal.

Drift Control with Changeover Costs

We model the problem of managing capacity in a build-to-order environment as a Brownian drift control problem and seek a policy that minimizes the long-term average cost. We assume the controller can, at some cost, shift the processing rate among a finite set of alternatives, for example by adding or removing staff, increasing or reducing the number of shifts, or opening or closing production lines. The controller incurs a cost for capacity per unit time and a delay cost that reflects the opportunity cost of revenue waiting to be recognized or the customer service impacts of delaying delivery of orders. Furthermore, he incurs a cost per unit to reject orders or idle resources as necessary to keep the workload of waiting orders within a prescribed range. We introduce a practical restriction on this problem, called the S-restricted Brownian control problem, and show how to model it via a structured linear program. We demonstrate that an optimal solution to the S-restricted problem can be found among a special class of policies called deterministic nonoverlapping control band policies. These results exploit apparently new relationships between complementary dual solutions and relative value functions that allow us to obtain a lower bound on the average cost of any nonanticipating policy for the problem, even without the S restriction. Under mild assumptions on the cost parameters, we show that our linear programming approach is asymptotically optimal for the unrestricted Brownian control problem in the sense that by appropriately selecting the S-restricted problem, we can ensure its solution is within an arbitrary finite tolerance of a lower bound on the average cost of any nonanticipating policy for the unrestricted Brownian control problem.

Matching 2-lattice polyhedra: finding a maximum vector

Abstract Matching 2-lattice polyhedra are a special class of lattice polyhedra that include network flow polyhedra, fractional matching polyhedra, matroid intersection polyhedra, etc. In this paper we develop a polynomial-time extreme point algorithm for finding a maximum cardinality vector in a matching 2-lattice polyhedron.