Commercial Optimization Solver Research Articles

Supply capacity acquisition and allocation with uncertain customer demands

We study a class of capacity acquisition and assignment problems with stochastic customer demands often found in operations planning contexts. In this setting, a supplier utilizes a set of distinct facilities to satisfy the demands of different customers or markets. Our model simultaneously assigns customers to each facility and determines the best capacity level to operate or install at each facility. We propose a branch-and-price solution approach for this new class of stochastic assignment and capacity planning problems. For problem instances in which capacity levels must fall between some pre-specified limits, we offer a tailored solution approach that reduces solution time by nearly 80% over an alternative approach using a combination of commercial nonlinear optimization solvers. We have also developed a heuristic solution approach that consistently provides optimal or near-optimal solutions, where solutions within 0.01% of optimality are found on average without requiring a nonlinear optimization solver.

Satisfying market demands with delivery obligations or delivery charges

Consider a firm that receives deterministic, time-varying demands for its product. The firm has the flexibility to choose its customer base and receive market-specific revenues for its product. In addition to market revenue and market demand, the parameters of a negotiated contract can also play a role in market selection and fulfillment decisions. Two such parameters are (1) delivery obligation penalties and (2) delivery charges. In this paper, we determine optimal solution approaches for problem settings when either of these contract parameters is present. Our tailored solution techniques can solve each problem in a fraction of the time required by a state-of-the-art commercial optimization solver.

A bilevel flow model for hazmat transportation network design

In this work we consider the following hazmat transportation network design problem. A given set of hazmat shipments has to be shipped over a road transportation network in order to transport a given amount of hazardous materials from specific origin points to specific destination points, and we assume there are regional and local government authorities that want to regulate the hazmat transportations by imposing restrictions on the amount of hazmat traffic over the network links. In particular, the regional authority aims to minimize the total transport risk induced over the entire region in which the transportation network is embedded, while local authorities want the risk over their local jurisdictions to be the lowest possible, forcing the regional authority to assure also risk equity. We provide a linear bilevel programming formulation for this hazmat transportation network design problem that takes into account both total risk minimization and risk equity. We transform the bilevel model into a single-level mixed integer linear program by replacing the second level (follower) problem by its KKT conditions and by linearizing the complementary constraints, and then we solve the MIP problem with a commercial optimization solver. The optimal solution may not be stable, and we provide an approach for testing its stability and for evaluating the range of its solution values when it is not stable. Moreover, since the bilevel model is difficult to be solved optimally and its optimal solution may not be stable, we provide a heuristic algorithm for the bilevel model able to always find a stable solution. The proposed bilevel model and heuristic algorithm are experimented on real scenarios of an Italian regional network.

Generating strictly non-self-overlapping structured quadrilateral grids

In this paper, we present a BPM (Bézier patch mapping) algorithm which generates a strictly non-self-overlapping structured quadrilateral grid in a given four-sided planar region. Given four pieces of polynomial curves which enclose a simple region in the plane, the algorithm first constructs a Bézier patch which interpolates the four curves (as its four boundary curves), while the inner control points of its control grid remain unknown. In this paper, we show that, for the bijective condition to be satisfied, it is sufficient that the interior points satisfy a set of quadratic inequality equations. Exploiting this key result, we formulate the mapping algorithm as an optimization problem where the constraints are the bijective condition of the Bézier patch mapping (BPM), and the objective is to find out the best from all of the non-self-overlapping grids. Thus, commercial optimization solvers can be used to find the bijective mapping. If a solution to the optimization problems exists, then so does a solution to the mapping problem, and vice-versa. The BPM method is simple and intuitive, and some examples presented in this paper demonstrate its effectiveness.

Solving asymmetric variational inequalities via convex optimization

Using duality, we reformulate the asymmetric variational inequality (VI) problem over a conic region as an optimization problem. We give sufficient conditions for the convexity of this reformulation. We thereby identify a class of VIs that includes monotone affine VIs over polyhedra, which may be solved by commercial optimization solvers.

The design and implementation of OR/SM: A prototype integrated modeling environment

This paper describes the design, implementation, and interaction of the processes of OR/SM, a computerized modeling environment built on ORACLE Tools and Database (OR) using Structured Modeling (SM) as the conceptual framework. Some of the key features of OR/SM include: (a) interactive checking of model syntax and semantics; (b) automatic gen-eration of relational database tables for model data; (c) automatic generation of several reference documents; (d) automatic and interactive links to SAS, a powerful and widely used commercial statistical analysis software system and optimization solver; and (e) an interactive link to QS (Quantitative Systems) - a commercial software package for solving a wide range of operations management models.