Integer linear programming models that incorporate probabilistic and stochastic components represent one approach for capturing the stochastic nature of emergency medical service ambulance systems. This includes modeling non-deterministic call arrival and servicing rates and congestion in the ambulance network (i.e., ambulance unavailability). These models focus on maximizing the total population that can find an available ambulance within a set service time standard (s) with a probability of at least α%. In MALP the concept of local vehicle busyness estimates is introduced to estimate the availability of service in a neighborhood given the neighborhood’s level of demand and the number of ambulance vehicles located in the neighborhood. QMALP is an extension of MALP where queue-theory derived parameters are implemented in the MALP model framework in order to relax the assumption that the probability of different ambulances being busy are independent. Despite this considerable development, several concerns remained about MALP and QMALP, namely the districting assumption where its assumed that a neighborhood’s calls for service are served only by an ambulance in the area, that ambulances in a neighborhood only serve calls for service originating within the neighborhood, or that at least the flow of ambulance service to and from external neighborhoods was roughly equal. Questions have been raised about the validity of MALP and QMALP’s reliability estimates, that is, whether a neighborhood actually received α-reliable service.

To address these issues, we developed the Resource-Constrained Queue-based Maximum Availability Location Problem (RC-QMALP). This model is based on a location-allocation framework that (1) assigns workload from neighborhoods to ambulances located within s and ambulance idle capacity to neighborhoods and (2) includes additional constraints designed to help ensure the validity of the original MALP and QMALP constraints used to establish whether a neighborhood can find an available ambulance with α-reliability. We also implemented a secondary minsum objective that minimizes the average travel distance between ambulances and the neighborhoods they service while maintaining the priority of the MALP and QMALP coverage objective.

In this thesis, we validated RC-QMALP by comparing the reliable coverage levels predicted by the RC-QMALP to the ambulance system simulations that used the locational configurations suggested by the RC-QMALP. We found that MALP 2 and QMALP provided higher levels of reliable coverage and that RC-QMALP’s secondary objective has a negligible impact on system performance. However, RC-QMALP-based models provide more accurate estimates of reliable coverage and location solutions whose simulated reliable coverage performance was always within 5% of the optimal solution with the same system parameters (we tested 1,080 different model configurations). Our work suggests that (1) simulation models must be developed to handle the modeling assumptions that underlie location optimization models and that (2) service reliability location models should consider additional factors such as ambulance workloads (and their distribution).

The basis of this dissertation work stems from the fact that if one examines system route maps for many bus transit systems in U.S. cities, an interesting pattern emerges. Routes often utilize embedded loops to increase accessibility coverage of a system at the expense of adding a marginal amount of length to the overall path. Further, such routes frequently share a common corridor with respect to traveling in opposing directions, but they may depart spatially from each other in terms of direction. These departures in direction represent embedded loops that are traversed in only one direction. However, the literature has not explored this issue, and in fact, often discourages or outright prevents any loops from occurring whether they are loops that are traversed in both directions or traversed in only one direction. Furthermore, past research on covering path models has not accounted for travel in opposing directions, even when attempting to model transit lines. This is due in part to the roots of the covering path literature.

This dissertation presents an analysis of past work and from that defines several new problems that are ‘loop agnostic’ – that is, they neither prevent nor encourage the formation of loops in an optimal route, essentially a new class of covering path problems. Although several loop agnostic models are developed in this dissertation to better represent the maximal covering shortest path problem, these models only capture one aspect of loop use. In the classic Maximal Covering Shortest Path problem, it is assumed that its use in transit will be traversed in both directions. Further, the classic formulation prevents most loops from occurring. A new form of this model is developed that allows loops to be part of a solution, whenever such loops provide an improvement in the objective function value. This model is called “loop agnostic” as the model neither prevents nor requires loops to be used in a solution. This means that a loop can be present as part of the path, as an out-and-back path or a more complex loop which visits several other nodes before returning to a previously visited node, or even as a ‘lollipop’ shaped route attached to the origin node or the destination node. If one assumes that the covering path can be traversed both in the outbound and inbound directions (which past work has done), any loops that are present will be traversed in both directions and is what we refer to as a bi-directional loop. When addressing the question of bi-directionality in real world systems it is possible that a loop is traversed in only one direction. Such “uni-directional” loops are formed whenever inbound/outbound paths diverge and can be observed in many transit system maps, like those of Bozeman, MT; Eau Claire, WI; and San Luis Obispo, CA. This dissertation also proposes a new problem, the Bi-Directional MCSP, and formulates two new models that account for travel based upon inbound and outbound path directions which allows for the use of shared arcs and uni-directional loops as well as bi-directional loops.

This dissertation also presents results from the application of these new models as well as a new heuristic to a hypothetical test network as well as a real world network from Richardson/Garland, Texas. Results demonstrate that loops are present in many optimal solutions and that the route designs that utilize loop structures such as a ‘lollipop,’ ‘barbell,’ and ‘figure eight’ may well be superior to route designs that do not incorporate loops. This gives credence to the designs of virtually all transit systems in small and medium sized cities in the United States.

Dispersive strategies and outcomes are readily apparent in many geographic contexts. In particular, dispersive strategies can be seen in activities such as: the scattering of military missile silos and ammunition bunkers, center-pivot crop irrigation systems, location of parks, franchise store location, and territorial species den/nest locations. Spatial optimization models represent dispersion where selected facility locations are maximally "packed" or maximally "separated." The Anti-Covering Location Problem, in particular, is one in which a maximum number of facilities are located within a region such that each facility is separated by at least a minimum distance standard from all others. In this context, facilities are "dispersed" from each other through the use of the minimum separation standard. Solutions to this problem are called maximally "packed" as there exists no opportunity to add facilities without violating minimum separation standards.

The Anti-Covering Location Problem (ACLP) can be defined on a continuous space domain, or more commonly, using a finite set of discrete locations. In this dissertation, it is assumed that there exists a discrete set of sites, among which a number will be selected for facility locations, and that this general problem may represent a number of different problems ranging from habitat analysis to public policy analysis. The main objective of this dissertation is to propose a new and improved optimization model for the ACLP when applied to a discrete set of points on a Cartesian plane using a combination of separation conditions called core-and-wedge constraints. This model structure, by its very definition, demonstrates that all planar problems can be defined using at most seven clique constraints for each site. In addition, the use of an added set of facet constraints in reducing computational effort is explored.

Anti-covering location model solutions are maximally packed, providing an "optimistic" estimate of what may be possible in dispersing facilities. But, what if less than optimal sites are employed in a dispersive pattern. That is, to what extent can an optimal maximally packed configuration be disrupted? This possibility is explored through the development of a new model, called the Disruptive Anti-Covering location model.

The problem of transmission line corridor location can be considered, at best, a "wicked" public systems decision problem. It requires the consideration of numerous objectives while balancing the priorities of a variety of stakeholders, and designers should be prepared to develop diverse non-inferior route alternatives that must be defensible under the scrutiny of a public forum. Political elements aside, the underlying geographical computational problems that must be solved to provide a set of high quality alternatives are no less easy, as they require solving difficult spatial optimization problems on massive GIS terrain-based raster data sets.

Transmission line siting methodologies have previously been developed to guide designers in this endeavor, but close scrutiny of these methodologies show that there are many shortcomings with their approaches. The main goal of this dissertation is to take a fresh look at the process of corridor location, and develop a set of algorithms that compute path alternatives using a foundation of solid geographical theory in order to offer designers better tools for developing quality alternatives that consider the entire spectrum of viable solutions. And just as importantly, as data sets become increasingly massive and present challenging computational elements, it is important that algorithms be efficient and able to take advantage of parallel computing resources.

A common approach to simplify a problem with numerous objectives is to combine the cost layers into a composite a priori weighted single-objective raster grid. This dissertation examines new methods used for determining a spatially diverse set of near-optimal alternatives, and develops parallel computing techniques for brute-force near-optimal path enumeration, as well as more elegant methods that take advantage of the hierarchical structure of the underlying path-tree computation to select sets of spatially diverse near optimal paths.

Another approach for corridor location is to simultaneously consider all objectives to determine the set of Pareto-optimal solutions between the objectives. This amounts to solving a discrete multi-objective shortest path problem, which is considered to be NP-Hard for computing the full set of non-inferior solutions. Given the difficulty of solving for the complete Pareto-optimal set, this dissertation develops an approximation heuristic to compute path sets that are nearly exact-optimal in a fraction of the time when compared to exact algorithms. This method is then applied as an upper bound to an exact enumerative approach, resulting in significant performance speedups. But as analytic computing continues to moved toward distributed clusters, it is important to optimize algorithms to take full advantage parallel computing. To that extent, this dissertation develops a scalable parallel framework that efficiently solves for the supported/convex solutions of a biobjective shortest path problem. This framework is equally applicable to other biobjective network optimization problems, providing a powerful tool for solving the next generation of location analysis and geographical optimization models.

Case Presentation: A 26-year-old male presented to our emergency department for six days of right-sided facial myasthenia and parasthesias following a dental procedure using anesthetic nerve blocks.

Discussion: Iatrogenic cranial nerve VII neuropraxia, a peripheral nerve injury, is an uncommon complication of alveolar nerve blocks with few documented cases specifically due to dental anesthesia. Treatment usually involves use of oral corticosteroid and/or antiviral medications along with close follow-up in clinic with a neurologist and/or otolaryngologist.