David P. Morton

Stochastic Vehicle Routing with Random Travel Times

We consider stochastic vehicle routing problems on a network with random travel and service times. A fleet of one or more vehicles is available to be routed through the network to service each node. Two versions of the model are developed based on alternative objective functions. We provide bounds on optimal objective function values and conditions under which reductions to simpler models can be made. Our solution method embeds a branch-and-cut scheme within a Monte Carlo sampling-based procedure.

SOCRATES: A system for scheduling hydroelectric generation under uncertainty

The Pacific Gas and Electric Company, the largest investor-owned energy utility in the United States, obtains a significant fraction of its electric energy and capacity from hydrogeneration. Although hydro provides valuable flexibility, it is subject to usage limits and must be carefully scheduled. In addition, the amount of energy available from hydro varies widely from year to year, depending on precipitation and streamflows. Optimal scheduling of hydrogeneration, in coordination with other energy sources, is a stochastic problem of practical significance to PG&E. SOCRATES is a system for the optimal scheduling of PG&Es various energy sources over a one- to two-year horizon. This paper concentrates on the component of SOCRATES that schedules hydro. The core is a stochastic optimization model, solved using Benders decomposition. Additional components are streamflow forecasting models and a database containing hydrological information. The stochastic hydro scheduling module of SOCRATES is undergoing testing in the users environment, and we expect PG&E hydrologists and hydro schedulers to place progressively more reliance upon it.

Models for nuclear smuggling interdiction

We describe two stochastic network interdiction models for thwarting nuclear smuggling. In the first model, the smuggler travels through a transportation network on a path that maximizes the probability of evading detection, and the interdictor installs radiation sensors to minimize that evasion probability. The problem is stochastic because the smugglers origin-destination pair is known only through a probability distribution at the time when the sensors are installed. In this model, the smuggler knows the locations of all sensors and the interdictor and the smuggler “agree” on key network parameters, namely the probabilities the smuggler will be detected while traversing the arcs of the transportation network. Our second model differs in that the interdictor and smuggler can have differing perceptions of these network parameters. This model captures the case in which the smuggler is aware of only a subset of the sensor locations. For both models, we develop the important special case in which the sensors can only be installed at border crossings of a single country so that the resulting model is defined on a bipartite network. In this special case, a class of valid inequalities reduces the computation time for the identical-perceptions model.

Cut sharing for multistage stochastic linear programs with interstage dependency

Multistage stochastic programs with interstage independent random parameters have recourse functions that do not depend on the state of the system. Decomposition-based algorithms can exploit this structure by sharing cuts (outer-linearizations of the recourse function) among different scenario subproblems at the same stage. The ability to share cuts is necessary in practical implementations of algorithms that incorporate Monte Carlo sampling within the decomposition scheme. In this paper, we provide methodology for sharing cuts in decomposition algorithms for stochastic programs that satisfy certain interstage dependency models. These techniques enable sampling-based algorithms to handle a richer class of multistage problems, and may also be used to accelerate the convergence of exact decomposition algorithms.

Optimizing Military Airlift

We describe a large-scale linear programming model for optimizing strategic (intercontinental) airlift capability. The model routes cargo and passengers through a specified transportation network with a given fleet of aircraft subject to many physical and policy constraints. The time-dynamic model captures a significant number of the important aspects of an airlift system in a large-scale military deployment, including aerial refueling, tactical (intracontinental) aircraft shuttles, and constraints based on crew availability. The model is designed to provide insight into issues associated with designing and operating an airlift system. We describe analyses for the U.S. Air Force system concerning fleet modernization and concerning the allocation of resources that affect the processing capacity of airfields.

A Sequential Sampling Procedure for Stochastic Programming

We develop a sequential sampling procedure for a class of stochastic programs. We assume that a sequence of feasible solutions with an optimal limit point is given as input to our procedure. Such a sequence can be generated by solving a series of sampling problems with increasing sample size, or it can be found by any other viable method. Our procedure estimates the optimality gap of a candidate solution from this sequence. If the point estimate of the optimality gap is sufficiently small according to our termination criterion, then we stop. Otherwise, we repeat with the next candidate solution from the sequence under an increased sample size. We provide conditions under which this procedure (i) terminates with probability one and (ii) terminates with a solution that has a small optimality gap with a prespecified probability.

An Enhanced Decomposition Algorithm for Multistage Stochastic Hydroelectric Scheduling

Handling uncertainty in natural inflow is an important part of a hydroelectric scheduling model. In a stochastic programming formulation, natural inflow may be modeled as a random vector with known distribution, but the size of the resulting mathematical program can be formidable. Decomposition-based algorithms take advantage of special structure and provide an attractive approach to such problems. We develop an enhanced Benders decomposition algorithm for solving multistage stochastic linear programs. The enhancements include warm start basis selection, preliminary cut generation, the multicut procedure, and decision tree traversing strategies. Computational results are presented for a collection of stochastic hydroelectric scheduling problems.

A stochastic program for interdicting smuggled nuclear material

This paper describes a stochastic network interdiction model for identifying locations for installing detectors sensitive to nuclear material. A nuclear material smuggler selects a path through a transportation network that maximizes the probability of avoiding detection An interdictor installs sensors to minimize that maximum probability. This problem is formulated as a bi-level stochastic mixed-integer program. The program is stochastic because the evader’s origin and destination are unknown at the time the detectors are installed. The model is reformulated as a two-stage stochastic mixed-integer program with recourse and is shown to be strongly NP-Hard. We describe an application of our model to help strengthen the overall capability of preventing the illicit trafficking of nuclear materials.

On a stochastic knapsack problem and generalizations

We consider an integer stochastic knapsack problem (SKP) where the weight of each item is deterministic, but the vector of returns for the items is random with known distribution. The objective is to maximize the probability that a total return threshold is met or exceeded. We study several solution approaches. Exact procedures, based on dynamic programming (DP) and integer programming (IP), are developed for returns that are independent normal random variables with integral means and variances. Computation indicates that the DP is significantly faster than the most efficient algorithm to date. The IP is less efficient, but is applicable to more general stochastic IPs with independent normal returns. We also develop a Monte Carlo approximation procedure to solve SKPs with general distributions on the random returns. This method utilizes upper- and lower-bound estimators on the true optimal solution value in order to construct a confidence interval on the optimality gap of a candidate solution.

Geometric optimization of radiant enclosures containing specular surfaces

This paper presents an optimization methodology for designing radiant enclosures containing specularly-reflecting surfaces. The optimization process works by making intelligent perturbations to the enclosure geometry at each design iteration using specialized numerical algorithms. This procedure requires far less time than the forward ‘‘trial-anderror’’ design methodology, and the final solution is near optimal. The radiant enclosure is analyzed using a Monte Carlo technique based on exchange factors, and the design is optimized using the Kiefer-Wolfowitz method. The optimization design methodology is demonstrated by solving two industrially-relevant design problems involving twodimensional enclosures that contain specular surfaces. @DOI: 10.1115/1.1599369#