Suvrajeet Sen

An Introductory Tutorial on Stochastic Linear Programming Models

Linear programming is a fundamental planning tool. It is often difficult to precisely estimate or forecast certain critical data elements of the linear program. In such cases, it is necessary to address the impact of uncertainty during the planning process. We discuss a variety of LP-based models that can be used for planning under uncertainty. In all cases, we begin with a deterministic LP model and show how it can be adapted to include the impact of uncertainty. We present models that range from simple recourse policies to more general two-stage and multistage SLP formulations. We also include a discussion of probabilistic constraints. We illustrate the various models using examples taken from the literature. The examples involve models developed for airline yield management, telecommunications, flood control, and production planning.

Network planning with random demand

We study a planning problem associated with networks for private line services. In these networks, demands are known to exhibit considerable variability, and as such, they should be treated as random variables. The proposed planning model is a two-stage stochastic linear program (SLP) with recourse. Due to the enormous size of the deterministic equivalent, we choose a sampling based algorithm calledstochastic decomposition (SD). For very large-scale SLPs, such as the ones solved in this application, SD provides an effective methodology. The model presented in this paper is validated by using a detailed simulation of the network. We report results with a network that has 86 demand pairs, 89 links and 706 potential routes.

CONTROLLED OPTIMIZATION OF PHASES AT AN INTERSECTION

This paper presents a general purpose algorithm for real-time traffic control at an intersection. Our methodology, based on dynamic programming, allows optimization of a variety of performance indices such as delay, stops and queue lengths. Furthermore, optimal phase sequencing is a direct by-product of this new approach. These features make the new methodology a powerful tool for intersection control. We demonstrate the usefulness of the approach by a simulation experiment in which our intersection control algorithm is interfaced with a well established simulation package called TRAF-NETSIM. Our study compares the controlled optimization of phases methodology with fully-actuated as well as semi-actuated control. We show that consistent reductions in delay may be possible by adopting the new algorithm.

A Branch-and-Price Algorithm for Multistage Stochastic Integer Programming with Application to Stochastic Batch-Sizing Problems

In this paper, we present a branch-and-price method to solve special structured multistage stochastic integer programming problems. We validate our method on two different versions of a multistage stochastic batch-sizing problem (SBSP). One version adopts a recourse formulation, and the other is based on probabilistic constraints. Our algorithmic approach is applicable to both formulations. Our computational results suggest that both classes of problems can be solved using relatively few nodes of a branch-and-price tree. The success of our approach calls for extensions in methodology as well as applications.

The C 3 Theorem and a D 2 Algorithm for Large Scale Stochastic Mixed-Integer Programming: Set Convexification

This paper considers the two-stage stochastic integer programming problem, with an emphasis on instances in which integer variables appear in the second stage. Drawing heavily on the theory of disjunctive programming, we characterize convexifications of the second stage problem and develop a decomposition-based algorithm for the solution of such problems. In particular, we verify that problems with fixed recourse are characterized by scenario-dependent second stage convexifications that have a great deal in common. We refer to this characterization as the C3 (Common Cut Coefficients) Theorem. Based on the C3 Theorem, we develop a decomposition algorithm which we refer to as Disjunctive Decomposition (D2). In this new class of algorithms, we work with master and subproblems that result from convexifications of two coupled disjunctive programs. We show that when the second stage consists of 0-1 MILP problems, we can obtain accurate second stage objective function estimates after finitely many steps. This result implies the convergence of the D2 algorithm.

A Mean-Variance Model for Route Guidance in Advanced Traveler Information Systems

Traditional models of route generation are based on choosing routes that minimize expected travel-time between origin and destination. Such approaches do not account for the fact that travelers often incorporate travel-time variability within their decision making. Thus, a route with lower travel-time variability is preferred by some travelers, even if such a route is not one with the lowest mean travel-time. Such traveler behavior is best captured by a multiobjective model in which the choice of a route is based on the mean as well as the variance of the path travel-time. Our route-planning model is intended to help travelers make choices that reflect their decision-making process better. We formulate a network flow multiobjective model in which one of the objectives (expectation) is linear, whereas, the other (variance) is quadratic. In order to present the user with a series of options, we solve a series of parametric 0-1 quadratic integer programs. By utilizing the network structure of the problem, we devise an effective algorithm in which the 0-1 quadratic program is solved by using a continuous relaxation together with an enumeration of some selected paths. Finally, we note that the data requirements for the model can be easily satisfied in practice.

A Stochastic Programming Approach to Power Portfolio Optimization

The DASH model for Power Portfolio Optimization provides a tool which helps decision-makers coordinate production decisions with opportunities in the wholesale power market. The methodology is based on a stochastic programming model which selects portfolio positions that perform well on a variety of scenarios generated through statistical modeling and optimization. When compared with a commonly used fixed-mix policy, our experiments demonstrate that the DASH model provides significant advantages over several fixed-mix policies.

The Scenario Generation Algorithm for Multistage Stochastic Linear Programming

A multistage stochastic linear program (MSLP) is a model of sequential stochastic optimization where the objective and constraints are linear. When any of the random variables used in the MSLP are continuous, the problem is infinite dimensional. To numerically tackle such a problem, we usually replace it with a finite-dimensional approximation. Even when all the random variables have finite support, the problem is often computationally intractable and must be approximated by a problem of smaller dimension. One of the primary challenges in the field of stochastic programming deals with discovering effective ways to evaluate the importance of scenarios and to use that information to trim the scenario tree in such a way that the solution to the smaller optimization problem is not much different from the problem stated with the original tree. The scenario generation (SG) algorithm proposed in this paper is a finite-element method that addresses this problem for the class of MSLP with random right-hand sides.

The Million-Variable March for Stochastic Combinatorial Optimization

Combinatorial optimization problems have applications in a variety of sciences and engineering. In the presence of data uncertainty, these problems lead to stochastic combinatorial optimization problems which result in very large scale combinatorial optimization problems. In this paper, we report on the solution of some of the largest stochastic combinatorial optimization problems consisting of over a million binary variables. While the methodology is quite general, the specific application with which we conduct our experiments arises in stochastic server location problems. The main observation is that stochastic combinatorial optimization problems are comprised of loosely coupled subsystems. By taking advantage of the loosely coupled structure, we show that decomposition-coordination methods provide highly effective algorithms, and surpass the scalability of even the most efficiently implemented backtracking search algorithms.

Relaxations for probabilistically constrained programs with discrete random variables

We consider mathematical programs with probabilistic constraints in which the random variables are discrete. In general, the feasible region associated with such problems is nonconvex. We use methods of disjunctive programming to approximate the convex hull of the feasible region. For a particular disjunctive set implied by the probabilistic constraint, we characterize the set of all valid inequalities as well as the facets of the convex hull of the given disjunctive set. These may be used within relaxation methods, especially for combinatorial optimization problems.