Alejandro Toriello

A Dynamic Traveling Salesman Problem with Stochastic Arc Costs

We propose a dynamic traveling salesman problem TSP with stochastic arc costs motivated by applications, such as dynamic vehicle routing, in which the cost of a decision is known only probabilistically beforehand but is revealed dynamically before the decision is executed. We formulate this as a dynamic program DP and compare it to static counterparts to demonstrate the advantage of the dynamic paradigm over an a priori approach. We then apply approximate linear programming ALP to overcome the DPs curse of dimensionality, obtain a semi-infinite linear programming lower bound, and discuss its tractability. We also analyze a rollout version of the price-directed policy implied by our ALP and derive worst-case guarantees for its performance. Our computational study demonstrates the quality of a heuristically modified rollout policy using a computationally effective a posteriori bound.

Fitting piecewise linear continuous functions

We consider the problem of fitting a continuous piecewise linear function to a finite set of data points, modeled as a mathematical program with convex objective. We review some fitting problems that can be modeled as convex programs, and then introduce mixed-binary generalizations that allow variability in the regions defining the best-fit function’s domain. We also study the additional constraints required to impose convexity on the best-fit function.

Fixed-Charge Transportation with Product Blending

Numerous planning models within the chemical, petroleum, and process industries involve coordinating the movement of raw materials in distribution networks so they can be blended into final products. The uncapacitated fixed-charge transportation problem with blending (FCTPwB) studied in this paper captures a core structure encountered in many of these environments. We model the FCTPwB as a mixed-integer linear program, and we derive two classes of facets, both exponential in size, for the convex hull of solutions for the problem with a single consumer and show that they can be separated in polynomial time. Furthermore, we prove that, in certain situations, these classes of facets along with the continuous relaxation of the original constraints yield a description of the convex hull. Finally, we present a computational study that demonstrates that these classes of facets are effective in reducing the integrality gap and solution time for more general instances of the FCTPwB with arc capacities and multiple consumers.

The One-Dimensional Dynamic Dispatch Waves Problem

We study same-day delivery systems by formulating the dynamic dispatch waves problem (DDWP), which models a depot where delivery requests arrive dynamically throughout a service day. At any dispatch epoch (wave), the information available to the decision maker is (1) a set of known, open requests that remain unfulfilled, and (2) a set of potential requests that may arrive later in the service day. At each wave, the decision maker decides whether or not to dispatch a vehicle, and if so, which subset of open requests to serve, with the objective of minimizing expected vehicle operating costs and penalties for unserved requests. We consider the DDWP with a single delivery vehicle and request destinations on a line, where vehicle operating times and costs depend only on the distance between points. We propose an efficient dynamic programming approach for the deterministic variant, and leverage it to design an optimal a priori policy with predetermined routes for the stochastic case. We then show that fully dy...

Optimal toll design: a lower bound framework for the asymmetric traveling salesman problem

We propose a framework of lower bounds for the asymmetric traveling salesman problem (TSP) based on approximating the dynamic programming formulation with different basis vector sets. We discuss how several well-known TSP lower bounds correspond to intuitive basis vector choices and give an economic interpretation wherein the salesman must pay tolls as he travels between cities. We then introduce an exact reformulation that generates a family of successively tighter lower bounds, all solvable in polynomial time. We show that the base member of this family yields a bound greater than or equal to the well-known Held-Karp bound, obtained by solving the linear programming relaxation of the TSP’s integer programming arc-based formulation.

Evaluation of Transportation Practices in the California Cut Flower Industry

In the past two decades, California’s share of the national cut flower market has decreased from 64 percent to 20 percent. California growers’ largest competitors are South American growers; Colombia controls 75 percent of the US market. South American growers have several competitive advantages, including the favorable trucking rates they enjoy by consolidating all shipments in Miami, Florida, prior to US distribution. This paper evaluates the California cut flower industry’s current transportation practices and investigates the feasibility and cost of establishing a shipping consolidation center in Oxnard, California. Applying a simple inventory management policy, we estimate a 35 percent system-wide transportation cost decrease of

Strategic Modeling of the Pediatric Nurse Practitioner Workforce

20 million per year if all California cut flower growers participate in the consolidation center. The California Cut Flower Commission incorporated our findings into an application for federal funds from the US Department of Transportation to construct a new flower transportation and logistics center in California. The state’s flower growers are also searching for alternative ways to cooperatively fund a consolidation center.

Technical Note—On Traveling Salesman Games with Asymmetric Costs

OBJECTIVE: To assess the current pediatric nurse practitioner (PNP) workforce and to investigate the impact of potential policy changes to address forecasted shortages. METHODS: We modeled the admission of students into nursing bachelor’s programs and followed them through advanced clinical programs. Prediction models were combined with optimal decision-making to determine best-case scenario admission levels. We computed 2 measures: (1) the absolute shortage and (2) the expected number of years until the PNP workforce will be able to fully satisfy PNP demand (ie, self-sufficiency). RESULTS: There is a forecasted shortage of PNPs in the workforce over the next 13 years. Under the best-case scenario, it would take at least 13 years for the workforce to fully satisfy demand. Our analysis of potential policy changes revealed that increasing the specialization rate for PNPs by 4% would decrease the number of years required until there are enough PNPs from 13 years to 5 years. Increasing the certification examination passing rate to 96% from the current average of 86.9% would lead to self-sufficiency in 11 years. In addition, increasing the annual growth rate of master’s programs to 36% from the current maximum of 10.7% would result in self-sufficiency in 5 years. CONCLUSIONS: Current forecasts of demand for PNPs indicate that the current workforce will be incapable of satisfying the growing demand. Policy changes can result in a reduction in the expected shortage and potentially improve access to care for pediatric patients.

Approximating the stability region for binary mixed-integer programs

We consider cooperative traveling salesman games with nonnegative asymmetric costs satisfying the triangle inequality. We construct a stable cost allocation with budget balance guarantee equal to the Held-Karp integrality gap for the asymmetric traveling salesman problem, using the parsimonious property and a previously unknown connection to linear production games. We also show that our techniques extend to larger classes of network design games. We then provide a simple example showing that our cost allocation does not necessarily achieve the best possible budget balance guarantee, even among cost allocations stable for the game defined by the Held-Karp relaxation, and discuss its implications on future work on traveling salesman games.

The Dynamic Dispatch Waves Problem for same-day delivery

The stability region of a solution is the polyhedral set of objective coefficients for which the solution is optimal. It provides valuable information for sensitivity analysis and re-optimization. An exact description of it may require an exponential number of inequalities. We develop polyhedral inner and outer approximations of linear size.