In this paper we introduce and solve a generalization of the classic average cost Brownian contro... more In this paper we introduce and solve a generalization of the classic average cost Brownian control problem in which a system manager dynamically controls the drift rate of a diffusion process X. At each instant, the system manager chooses the drift rate from a pair {u, v} of available rates and can invoke instantaneous controls either to keep X from falling or to keep it from rising. The objective is to minimize the long-run average cost consisting of holding or delay costs, processing costs, costs for invoking instantaneous controls, and fixed costs for changing the drift rate. We provide necessary and sufficient conditions on the cost parameters to ensure the problem admits a finite optimal solution. When it does, a simple control band policy specifying economic buffer sizes (α, Ω) and up to two switching points is optimal. The controller should invoke instantaneous controls to keep X in the interval (α, Ω). A policy with no switching points relies on a single drift rate exclusive...
We develop heuristics for a problem that models the static balancing of turbine fans: load point ... more We develop heuristics for a problem that models the static balancing of turbine fans: load point masses at regularly spaced positions on the periphery of a circle so that the residual unbalance about the center—which corresponds to the axis of rotation of the fan—is as small as possible. We give worst-case guarantees for our heuristics in terms of residual unbalance. For the case of an even number of blades, we show that one of our heuristics provides the same worst-case guarantee (with respect to the ideal of perfect balance) as does total enumeration. Furthermore, computational tests show that our heuristics are orders of magnitude faster and not far from optimum on average.
We study a two-machine flowshop in which all processing times are independently and identically d... more 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.
When a manufacturer places repeated orders with a supplier to meet changing production requiremen... more 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 expe...
This is the first in a series of papers that explores a class of polyhedra we call 2-lattice poly... more This is the first in a series of papers that explores a class of polyhedra we call 2-lattice polyhedra. 2-Lattice polyhedra are a special class of lattice polyhedra that include network flow polyhedra, fractional matching polyhedra, matroid intersection polyhedra, the intersection of two polymatroids, etc. In this paper we show that the maximum sum of components of a vector in a 2-lattice polyhderon is equal to the minimum capacity of a cover for the polyhedron. For special classes of 2-lattice polyhedra, called matching 2-lattice polyhedra, that include all of the mentioned special cases except the intersection of two polymatroids, we characterize the largest member in the family of minimum covers in terms of the maximum “cardinality” vectors in the polyhedron. In fact, we show that this same characterization arises from considering only the extreme maximum cardinality vectors. This characterization is at the heart of our extreme point algorithm [3] for finding a maximum cardinalit...
We model the problem of managing capacity in a build-to-order environment as a Brownian drift con... more 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 by, for example, 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 $Ss$-restricted Brownian control problem, and show how to model it via a structured linear program. We demonstrate that an optimal solution to the $Ss$-restricted proble...
This is the first in a series of papers that explores a class of polyhedra we call 2-lattice poly... more This is the first in a series of papers that explores a class of polyhedra we call 2-lattice polyhedra. 2-Lattice polyhedra are a special class of lattice polyhedra that include network flow polyhedra, fractional matching polyhedra, matroid intersection polyhedra, the intersection of two polymatroids, etc. In this paper we show that the maximum sum of components of a vector in a 2-lattice polyhderon is equal to the minimum capacity of a cover for the polyhedron. For special classes of 2-lattice polyhedra, called matching 2-lattice polyhedra, that include all of the mentioned special cases except the intersection of two polymatroids, we characterize the largest member in the family of minimum covers in terms of the maximum "cardinality" vectors in the polyhedron. In fact, we show that this same characterization arises from considering only the extreme maximum cardinality vectors. This characterization is at the heart of our extreme point algorithm [3] for finding a maximum ...
International Series in Operations Research & Management Science, 2011
ABSTRACT Auto manufacturers today face many challenges: The industry is plagued with excess capac... more ABSTRACT Auto manufacturers today face many challenges: The industry is plagued with excess capacity that drives down prices, international competitors are seizing share at both ends of the market and consumers are well informed about options and prices. All these factors combine to heighten competitive pressures, squeeze margins, and leave manufacturers struggling to increase revenues and market share.
In this paper we introduce and solve a generalization of the classic average cost Brownian contro... more In this paper we introduce and solve a generalization of the classic average cost Brownian control problem in which a system manager dynamically controls the drift rate of a diffusion process X. At each instant, the system manager chooses the drift rate from a pair {u, v} of available rates and can invoke instantaneous controls either to keep X from falling or to keep it from rising. The objective is to minimize the long-run average cost consisting of holding or delay costs, processing costs, costs for invoking instantaneous controls, and fixed costs for changing the drift rate. We provide necessary and sufficient conditions on the cost parameters to ensure the problem admits a finite optimal solution. When it does, a simple control band policy specifying economic buffer sizes (α, Ω) and up to two switching points is optimal. The controller should invoke instantaneous controls to keep X in the interval (α, Ω). A policy with no switching points relies on a single drift rate exclusive...
We develop heuristics for a problem that models the static balancing of turbine fans: load point ... more We develop heuristics for a problem that models the static balancing of turbine fans: load point masses at regularly spaced positions on the periphery of a circle so that the residual unbalance about the center—which corresponds to the axis of rotation of the fan—is as small as possible. We give worst-case guarantees for our heuristics in terms of residual unbalance. For the case of an even number of blades, we show that one of our heuristics provides the same worst-case guarantee (with respect to the ideal of perfect balance) as does total enumeration. Furthermore, computational tests show that our heuristics are orders of magnitude faster and not far from optimum on average.
We study a two-machine flowshop in which all processing times are independently and identically d... more 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.
When a manufacturer places repeated orders with a supplier to meet changing production requiremen... more 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 expe...
This is the first in a series of papers that explores a class of polyhedra we call 2-lattice poly... more This is the first in a series of papers that explores a class of polyhedra we call 2-lattice polyhedra. 2-Lattice polyhedra are a special class of lattice polyhedra that include network flow polyhedra, fractional matching polyhedra, matroid intersection polyhedra, the intersection of two polymatroids, etc. In this paper we show that the maximum sum of components of a vector in a 2-lattice polyhderon is equal to the minimum capacity of a cover for the polyhedron. For special classes of 2-lattice polyhedra, called matching 2-lattice polyhedra, that include all of the mentioned special cases except the intersection of two polymatroids, we characterize the largest member in the family of minimum covers in terms of the maximum “cardinality” vectors in the polyhedron. In fact, we show that this same characterization arises from considering only the extreme maximum cardinality vectors. This characterization is at the heart of our extreme point algorithm [3] for finding a maximum cardinalit...
We model the problem of managing capacity in a build-to-order environment as a Brownian drift con... more 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 by, for example, 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 $Ss$-restricted Brownian control problem, and show how to model it via a structured linear program. We demonstrate that an optimal solution to the $Ss$-restricted proble...
This is the first in a series of papers that explores a class of polyhedra we call 2-lattice poly... more This is the first in a series of papers that explores a class of polyhedra we call 2-lattice polyhedra. 2-Lattice polyhedra are a special class of lattice polyhedra that include network flow polyhedra, fractional matching polyhedra, matroid intersection polyhedra, the intersection of two polymatroids, etc. In this paper we show that the maximum sum of components of a vector in a 2-lattice polyhderon is equal to the minimum capacity of a cover for the polyhedron. For special classes of 2-lattice polyhedra, called matching 2-lattice polyhedra, that include all of the mentioned special cases except the intersection of two polymatroids, we characterize the largest member in the family of minimum covers in terms of the maximum "cardinality" vectors in the polyhedron. In fact, we show that this same characterization arises from considering only the extreme maximum cardinality vectors. This characterization is at the heart of our extreme point algorithm [3] for finding a maximum ...
International Series in Operations Research & Management Science, 2011
ABSTRACT Auto manufacturers today face many challenges: The industry is plagued with excess capac... more ABSTRACT Auto manufacturers today face many challenges: The industry is plagued with excess capacity that drives down prices, international competitors are seizing share at both ends of the market and consumers are well informed about options and prices. All these factors combine to heighten competitive pressures, squeeze margins, and leave manufacturers struggling to increase revenues and market share.
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Papers by John Vande Vate