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Opioid Use Disorder (OUD) has reached an epidemic level in the US. Diversion of unused prescription opioids to secondary users and black market significantly contributes to the abuse and misuse of these highly addictive drugs, leading to the increased risk of OUD and accidental opioid overdose within communities. Hence, it is critical to design effective strategies to reduce the non-medical use of opioids that can occur via diversion at the patient level. In this paper, we aim to address this critical public health problem by designing strategies for the return and safe disposal of unused prescription opioids. We propose a data-driven optimization framework to determine the optimal incentive disbursement plans and locations of easily accessible opioid disposal kiosks to motivate prescription opioid users of diverse profiles in returning their unused opioids. We develop a Mixed-Integer Non-Linear Programming (MINLP) model to solve the decision problem, followed by a reformulation scheme using Benders Decomposition that results in a computationally efficient solution. We present a case study to show the benefits and usability of the model using a dataset created from Massachusetts All Payer Claims Data (MA APCD). Our proposed model allows the policymakers to estimate and include a penalty cost considering the economic and healthcare burden associated with prescription opioid diversion. Our numerical experiments demonstrate the ability of model and usefulness in determining optimal locations of opioid disposal kiosks and incentive disbursement plans for maximizing the disposal of unused opioids. The proposed optimization framework offers various trade-off strategies that can help government agencies design pragmatic policies for reducing the diversion of unused prescription opioids.

Interior Point Methods are widely used to solve Linear Programming problems. In this work, we present two primal Affine Scaling algorithms to achieve faster convergence in solving Linear Programming problems. In the first algorithm, we integrate Nesterov’s restarting strategy in the primal Affine Scaling method with an extra parameter, which in turn generalizes the original primal Affine Scaling method. We provide the proof of convergence for the proposed generalized algorithm considering long step size. We also provide the proof of convergence for the primal and dual sequence without the degeneracy assumption. This convergence result generalizes the original convergence result for the Affine Scaling methods and it gives us hints about the existence of a new family of methods. Then, we introduce a second algorithm to accelerate the convergence rate of the generalized algorithm by integrating a non-linear series transformation technique. Our numerical results show that the proposed algorithms outperform the original primal Affine Scaling method.

Healthcare facility location decisions are of great importance due to their impact on direct and social cost of people's well-being in a region. Optimal location decisions considering only current demand may become suboptimal as demand distribution changes. Considering future demand realisations in the decision making process can ensure long-term optimality. We present three mathematical models which follow grid-based location approach, and consider current and future demands in providing optimal location-allocation decisions. The first model considers allocations of present and future patients only to the nearest facilities. The second model allows patients to travel to facilities within allowable distance. The third model allows allocation of patients from one location to multiple facilities. The models are implemented with AMPL and numerical instances are solved with the CPLEX solver. Results show that the models are capable of solving medium size problems and the third model performs better in providing high quality solutions.

Grid-based location problems (GBLPs) can be used to solve location problems in business, engineering, resource exploitation, and even in the field of medical sciences. To solve these decision problems, an integer linear programming (ILP) model is designed and developed to provide the optimal solution for GBLPs considering fixed cost criteria. Preliminary results show that the ILP model is efficient in solving small to moderate-sized problems. However, this ILP model becomes intractable in solving large-scale instances. Therefore, a decomposition heuristic is proposed to solve these large-scale GBLPs, which demonstrates significant reduction of solution runtimes. To benchmark the proposed heuristic, results are compared with the exact solution via ILP. The experimental results show that the proposed method significantly outperforms the exact method in runtime with minimal (and in most cases, no) loss of optimality.

Selecting optimal location is a key decision problem in business and engineering. This research focuses to develop mathematical models for a special type of location problems called grid-based location problems. It uses a real-world problem of placing lights in a park to minimize the amount of darkness and excess supply. The non-linear nature of the supply function (arising from the light physics) and heterogeneous demand distribution make this decision problem truly intractable to solve. We develop ILP models that are designed to provide the optimal solution for the light post problem: the total number of light posts, the location of each light post, and their capacities (i.e., brightness). Finally, the ILP models are implemented within a standard modeling language and solved with the CPLEX solver. Results show that the ILP models are quite efficient in solving moderately sized problems with a very small optimality gap.

In multi-stage project investment decision-making with uncertainty, risk mitigation plays a vital role. The return on investment (ROI) that will be realised in making a particular decision quite often carries a high degree of uncertainty, with an increased number of competing investors entering to the market every day. In this research, our objective is to develop a technique for a multi-stage project investment decision problem that deals with uncertainty in ROI and complex interrelated state transition dynamics. We do this by formulating our problem as an infinite horizon stochastic dynamic programming (IHSDP) problem and solve it to maximise the total return over an infinite time horizon. We have implemented our solution to the project investment decision problem in a simple case study using three well-known stochastic dynamic programming algorithms. Our simulation results show that the IHSDP algorithms are useful in making optimum investment decisions in an uncertain business environment.