Multi-resource allocation and care sequence assignment in patient management: a stochastic programming approach.

To mitigate outpatient care delivery inefficiencies induced by resource shortages and demand heterogeneity, this paper focuses on the problem of allocating and sequencing multiple medical resources so that patients scheduled for clinical care can experience efficient and coordinated care with minimum total waiting time. We leverage highly granular location data on people and medical resources collected via Real-Time Location System technologies to identify dominant patient care pathways. A novel two-stage Stochastic Mixed Integer Linear Programming model is proposed to determine the optimal patient sequence based on the available resources according to the care pathways that minimize patients' expected total waiting time. The model incorporates the uncertainty in care activity duration via sample average approximation.We employ a Monte Carlo Optimization procedure to determine the appropriate sample size to obtain solutions that provide a good trade-off between approximation accuracy and computational time. Compared to the conventional deterministic model, our proposed model would significantly reduce waiting time for patients in the clinic by 60%, on average, with acceptable computational resource requirements and time complexity. In summary, this paper proposes a computationally efficient formulation for the multi-resource allocation and care sequence assignment optimization problem under uncertainty. It uses continuous assignment decision variables without timestamp and position indices, enabling the data-driven solution of problems with real-time allocation adjustment in a dynamic outpatient environment with complex clinical coordination constraints.

A two-stage stochastic optimization framework to allocate operating room capacity in publicly-funded hospitals under uncertainty.

Surgery demand is an uncertain parameter in addressing the problem of surgery block allocations, and its typical variability should be considered to ensure the feasibility of surgical planning. We develop two models, a stochastic recourse programming model and a two-stage stochastic optimization (SO) model with incorporated risk measure terms in the objective functions to determine a planning decision that is made to allocate surgical specialties to operating rooms (ORs). Our aim is to minimize the costs associated with postponements and unscheduled demands as well as the inefficient use of OR capacity. The results of these models are compared using a case of a real-life hospital to determine which model better copes with uncertainty. We propose a novel framework to transform the SO model based on its deterministic counterpart. Three SO models are proposed with respect to the variability and infeasibility of the measures of the objective function to encode the construction of the SO framework. The analysis of the experimental results demonstrates that the SO model offers better performance under a highly volatile demand environment than the recourse model. The originality of this work lies in its use of SO transformation framework and its development of stochastic models to address the problem of surgery capacity allocation based on a real case.

Practicable robust stochastic optimization under divergence measures with an application to equitable humanitarian response planning.

We seek to provide practicable approximations of the two-stage robust stochastic optimization model when its ambiguity set is constructed with an f-divergence radius. These models are known to be numerically challenging to various degrees, depending on the choice of the f-divergence function. The numerical challenges are even more pronounced under mixed-integer first-stage decisions. In this paper, we propose novel divergence functions that produce practicable robust counterparts, while maintaining versatility in modeling diverse ambiguity aversions. Our functions yield robust counterparts that have comparable numerical difficulties to their nominal problems. We also propose ways to use our divergences to mimic existing f-divergences without affecting the practicability. We implement our models in a realistic location-allocation model for humanitarian operations in Brazil. Our humanitarian model optimizes an effectiveness-equity trade-off, defined with a new utility function and a Gini mean difference coefficient. With the case study, we showcase (1) the significant improvement in practicability of the robust stochastic optimization counterparts with our proposed divergence functions compared to existing f-divergences, (2) the greater equity of humanitarian response that the objective function enforces and (3) the greater robustness to variations in probability estimations of the resulting plans when ambiguity is considered.

Capacity planning for effective cohorting of hemodialysis patients during the coronavirus pandemic: A case study.

Planning treatments of different types of patients have become challenging in hemodialysis clinics during the COVID-19 pandemic due to increased demands and uncertainties. In this study, we address capacity planning decisions of a hemodialysis clinic, located within a major public hospital in Istanbul, which serves both infected and uninfected patients during the COVID-19 pandemic with limited resources (i.e., dialysis machines). The clinic currently applies a 3-unit cohorting strategy to treat different types of patients (i.e., uninfected, infected, suspected) in separate units and at different times to mitigate the risk of infection spread risk. Accordingly, at the beginning of each week, the clinic needs to allocate the available dialysis machines to each unit that serves different patient cohorts. However, given the uncertainties in the number of different types of patients that will need dialysis each day, it is a challenge to determine which capacity configuration would minimize the overlapping treatment sessions of different cohorts over a week. We represent the uncertainties in the number of patients by a set of scenarios and present a stochastic programming approach to support capacity allocation decisions of the clinic. We present a case study based on the real-world patient data obtained from the hemodialysis clinic to illustrate the effectiveness of the proposed model. We also compare the performance of different cohorting strategies with three and two patient cohorts.

Design of control strategies to help prevent the spread of COVID-19 pandemic.

This paper proposes control strategies to allocate COVID-19 patients to screening facilities, health facilities, and quarantine facilities for minimizing the spread of the virus by these patients. To calculate the transmission rate, we propose a function that accounts for contact rate, duration of the contact, age structure of the population, susceptibility to infection, and the number of transmission events per contact. Moreover, the COVID-19 cases are divided into different groups according to the severity of their disease and are allocated to appropriate health facilities that provide care tailored to their needs. The multi-stage fuzzy stochastic programming approach is applied to cope with uncertainty, in which the probability associated with nodes of the scenario tree is treated as fuzzy variables. To handle the probabilistic model, we use a more flexible measure, M e measure, which allows decision-makers to adopt varying attitudes by assigning the optimistic-pessimistic parameter. This measure does not force decision-makers to hold extreme views and obtain the interval solution that provides further information in the fuzzy environment. We apply the proposed model to the case of Tehran, Iran. The results of this study indicate that assigning patients to appropriate medical centers improves the performance of the healthcare system. The result analysis highlights the impact of the demographic differences on virus transmission, and the older population has a greater influence on virus transmission than other age groups. Besides, the results indicate that behavioral changes in the population and their vaccination play a key role in curbing COVID-19 transmission.

Resource planning strategies for healthcare systems during a pandemic.

We study resource planning strategies, including the integrated healthcare resources' allocation and sharing as well as patients' transfer, to improve the response of health systems to massive increases in demand during epidemics and pandemics. Our study considers various types of patients and resources to provide access to patient care with minimum capacity extension. Adding new resources takes time that most patients don't have during pandemics. The number of patients requiring scarce healthcare resources is uncertain and dependent on the speed of the pandemic's transmission through a region. We develop a multi-stage stochastic program to optimize various strategies for planning limited and necessary healthcare resources. We simulate uncertain parameters by deploying an agent-based continuous-time stochastic model, and then capture the uncertainty by a forward scenario tree construction approach. Finally, we propose a data-driven rolling horizon procedure to facilitate decision-making in real-time, which mitigates some critical limitations of stochastic programming approaches and makes the resulting strategies implementable in practice. We use two different case studies related to COVID-19 to examine our optimization and simulation tools by extensive computational results. The results highlight these strategies can significantly improve patient access to care during pandemics; their significance will vary under different situations. Our methodology is not limited to the presented setting and can be employed in other service industries where urgent access matters.

Designing a sustainable-resilient-responsive supply chain network considering uncertainty in the COVID-19 era.

Effective supply chain management is crucial for economic growth, and sustainability is becoming a key consideration for large companies. COVID-19 has presented significant challenges to supply chains, making PCR testing a vital product during the pandemic. It detects the presence of the virus if you are infected at the time and detects fragments of the virus even after you are no longer infected. This paper proposes a multi-objective mathematical linear model to optimize a sustainable, resilient, and responsive supply chain for PCR diagnostic tests. The model aims to minimize costs, negative societal impact caused by shortages, and environmental impact, using a scenario-based approach with stochastic programming. The model is validated by investigating a real-life case study in one of Iran's high-risk supply chain areas. The proposed model is solved using the revised multi-choice goal programming method. Lastly, sensitivity analyses based on effective parameters are conducted to analyze the behavior of the developed Mixed-Integer Linear Programming. According to the results, not only is the model capable of balancing three objective functions, but it is also capable of providing resilient and responsive networks. To enhance the design of the supply chain network, this paper has considered various COVID-19 variants and their infectious rates, in contrast to prior studies that did not consider the variations in demand and societal impact exhibited by different virus variants.

An integrated chance constraints approach for optimal vaccination strategies under uncertainty for COVID-19.

Despite concerted efforts by health authorities worldwide to contain COVID-19, the SARS-CoV-2 virus has continued to spread and mutate into new variants with uncertain transmission characteristics. Therefore, there is a need for new data-driven models for determining optimal vaccination strategies that adapt to the new variants with their uncertain transmission characteristics. Motivated by this challenge, we derive an integrated chance constraints stochastic programming (ICC-SP) approach for finding vaccination strategies for epidemics that incorporates population demographics for any region of the world, uncertain disease transmission and vaccine efficacy. An optimal vaccination strategy specifies the proportion of individuals in a given household-type to vaccinate to bring the reproduction number to below one. The ICC-SP approach provides a quantitative method that allows to bound the expected excess of the reproduction number above one by an acceptable amount according to the decision-maker's level of risk. This new methodology involves a multi-community household based epidemiology model that uses census demographics data, vaccination status, age-related heterogeneity in disease susceptibility and infectivity, virus variants, and vaccine efficacy. The new methodology was tested on real data for seven neighboring counties in the United States state of Texas. The results are promising and show, among other findings, that vaccination strategies for controlling an outbreak should prioritize vaccinating certain household sizes as well as age groups with relatively high combined susceptibility and infectivity.

Designing a multi-objective closed-loop supply chain: a two-stage stochastic programming, method applied to the garment industry in Montréal, Canada.

The global population continues to grow, which expands demand for raw materials. Meanwhile, governments are developing circular economy strategies within cities and their industries. A circular economy utilizes refurbishing, reusing, remanufacturing, and repairing of products and materials. For companies, this involves to set targets and to rethink their supply chain. This paper seeks to model an exhaustive multi-echelon closed-loop supply chain (CLSC) network. This network functions within uncertainty, and the model optimizes three different objectives. The first objective function maximizes the network's profit; the second objective function minimizes network emissions. The last objective function maximizes job positions created by the network. Optimizing three contradicting objectives is a problem, so an augmented epsilon constraint method is applied to improve the model. Given the rise of fast fashion in developed countries, this model is used in the clothing industry in Montreal, Canada. The model includes three scenarios over five years with two types of products. The result shows the attractiveness of such a network for companies looking for profit, sustainability, and entrepreneurship in the garment industry.

Supply chain planning of vaccine and pharmaceutical clusters under uncertainty: The case of COVID-19.

As an abrupt epidemic occurs, healthcare systems are shocked by the surge in the number of susceptible patients' demands, and decision-makers mostly rely on their frame of reference for urgent decision-making. Many reports have declared the COVID-19 impediments to trading and global economic growth. This study aims to provide a mathematical model to support pharmaceutical supply chain planning during the COVID-19 epidemic. Additionally, it aims to offer new insights into hospital supply chain problems by unifying cold and non-cold chains and considering a wide range of pharmaceuticals and vaccines. This approach is unprecedented and includes an analysis of various pharmaceutical features such as temperature, shelf life, priority, and clustering. To propose a model for planning the pharmaceutical supply chains, a mixed-integer linear programming (MILP) model is used for a four-echelon supply chain design. This model aims to minimize the costs involved in the pharmaceutical supply chain by maintaining an acceptable service level. Also, this paper considers uncertainty as an intrinsic part of the problem and addresses it through the wait-and-see method. Furthermore, an unexplored unsupervised learning method in the realm of supply chain planning has been used to cluster the pharmaceuticals and the vaccines and its merits and drawbacks are proposed. A case of Tehran hospitals with real data has been used to show the model's capabilities, as well. Based on the obtained results, the proposed approach is able to reach the optimum service level in the COVID conditions while maintaining a reduced cost. The experiment illustrates that the hospitals' adjacency and emergency orders alleviated the service level significantly. The proposed MILP model has proven to be efficient in providing a practical intuition for decision-makers. The clustering technique reduced the size of the problem and the time required to solve the model considerably.

A Sharp Estimate on the Transient Time of Distributed Stochastic Gradient Descent.

This paper is concerned with minimizing the average of n cost functions over a network in which agents may communicate and exchange information with each other. We consider the setting where only noisy gradient information is available. To solve the problem, we study the distributed stochastic gradient descent (DSGD) method and perform a non-asymptotic convergence analysis. For strongly convex and smooth objective functions, in expectation, DSGD asymptotically achieves the optimal network independent convergence rate compared to centralized stochastic gradient descent (SGD). Our main contribution is to characterize the transient time needed for DSGD to approach the asymptotic convergence rate. Moreover, we construct a "hard" optimization problem that proves the sharpness of the obtained result. Numerical experiments demonstrate the tightness of the theoretical results.

Two-Stage Stochastic Optimization for the Pre-Position and Reconfiguration of Microgrid Defense Resources against Natural Disasters.

With the aggravation and evolution of global warming, natural disasters such as hurricanes occur more frequently, posing a great challenge to large-scale power systems. Therefore, the pre-position and reconfiguration of the microgrid defense resources by means of Mobile Energy Storage Vehicles (MEVs) and tie lines in damaged scenarios have attracted more and more attention. This paper proposes a novel two-stage optimization model with the consideration of MEVs and tie lines to minimize the shed loads and the outage duration of loads according to their proportional priorities. In the first stage, tie lines addition and MEVs pre-position are decided prior to a natural disaster; in the second stage, the switches of tie lines and original lines are operated and MEVs are allocated from staging locations to allocation nodes according to the specific damaged scenarios after the natural disaster strikes. The proposed load restoration method exploits the benefits of MEVs and ties lines by microgrid formation to pick up more critical loads. The progressive hedging algorithm is employed to solve the proposed scenario-based two-stage stochastic optimization problem. Finally, the effectiveness and superiority of the proposed model and applied algorithm are validated on an IEEE 33-bus test case.

Optimal configuration of low impact development practices for the management of urban runoff pollution under uncertainty.

The urbanization process has seen an accelerated increase in recent decades, leading to urban runoff pollution becoming more prominent. However, uncertainty of the pollution output and complexity of management systems have made controlling urban runoff pollution challenging. Therefore, it is necessary to propose advanced modeling methods for these challenges. This research presents an integrated urban runoff pollution management (IURPM) model for optimal configuration of low impact development (LID) practices under multiple uncertainties. The IURPM model combines the hybrid land-use prediction and improved pollution estimation models with interval parameter, stochastic parameter, and multi-objective programming. The proposed IURPM model can not only predict the output characteristics, but also provide optimal configuration schemes for the LID practices in the management of urban runoff pollution under multiple scenarios. In addition, uncertainties expressed as discrete intervals and probability density function in the management systems can be effectively addressed. A case study of the IURPM model was conducted in Dongguan City, South China. Results show that considerable amounts of urban runoff pollutants would export from Dongguan City by 2025. The export loads and pollution output flux per unit area would have significant spatial heterogeneity. The results further indicate that population size, gross domestic product, and regional area size are expected to play important roles in the pollution export, while impervious surface coverage and population density would likely have great influences on the output flux of urban runoff pollution. Based on the model findings, multiple LID practices should be adopted in Dongguan City to reduce the urban runoff pollution loads. Using the IURPM model, multiple LID implementation schemes can be obtained under different pollution reduction scenarios and significance levels, that can provide decision-making support for urban water environmental management, considering variations in the policymaker's decision-making preferences. This study demonstrates that the IURPM model can be applied to the optimal configuration of LID practices for the management of urban runoff pollution under uncertainty.

Optimal design of two-dimensional water trading based on risk aversion for sustainable development of Daguhe watershed, China.

Water related problems, including water scarcity and pollution, have become increasingly urgent challenges especially in arid and semiarid regions. Two-dimensional water trading (2DWT) mechanism has been designed to unify the quantity and quality of water for relieving the water crisis. This study aims to develop a risk aversion optimization-two dimensional water trading model (RAO-2DWTM) for planning the regional-scale water resources management system (RWMS). This is the first attempt on planning RWMS through risk aversion optimization within the two-dimensional water trading framework. RAO-2DWTM cannot only support in-depth analysis regarding the effect of decision maker's preferences on system risk in different trading scenarios, but also reflect the interaction between water right trading and effluent trading, as well as disclose the optimal scheme of water resource management under uncertainties. Twenty four scenarios associated with different trading scenarios and robust levels are analyzed. The optimization scheme under the optimal risk control level is determined based on TOPSIS. Results revealed that 2DWT would bring high benefit with reduced risk cost, water deficit and emissions, implying the effectiveness of 2DWT mechanism. The results also disclosed that risk aversion behavior can mitigate water scarcity and pollution, as well as reduce risk cost, but may lead to some losses of system benefit. Consequently, decision makers should make trade-offs between system benefit and risk in identifying desired trading schemes.

About the complexity of two-stage stochastic IPs.

We consider so called 2-stage stochastic integer programs (IPs) and their generalized form, so called multi-stage stochastic IPs. A 2-stage stochastic IP is an integer program of the form max { c T x ∣ A x = b , l ≤ x ≤ u , x ∈ Z s + n t } where the constraint matrix A ∈ Z r n × s + n t consists roughly of n repetitions of a matrix A ∈ Z r × s on the vertical line and n repetitions of a matrix B ∈ Z r × t on the diagonal. In this paper we improve upon an algorithmic result by Hemmecke and Schultz from 2003 [Hemmecke and Schultz, Math. Prog. 2003] to solve 2-stage stochastic IPs. The algorithm is based on the Graver augmentation framework where our main contribution is to give an explicit doubly exponential bound on the size of the augmenting steps. The previous bound for the size of the augmenting steps relied on non-constructive finiteness arguments from commutative algebra and therefore only an implicit bound was known that depends on parameters r, s, t and Δ , where Δ is the largest entry of the constraint matrix. Our new improved bound however is obtained by a novel theorem which argues about intersections of paths in a vector space. As a result of our new bound we obtain an algorithm to solve 2-stage stochastic IPs in time f ( r , s , Δ ) · poly ( n , t ) , where f is a doubly exponential function. To complement our result, we also prove a doubly exponential lower bound for the size of the augmenting steps.

Nutrient requirement equations for Indian goat by multiple regression analysis and least cost ration formulation using a linear and non-linear stochastic model.

An elaborate multiple regression analysis was done to arrive a nutrient requirement equation for goat including dry matter intake, DMI (kg/day), total digestible nutrient, TDN (g/day) and crude protein, CP (g/day) based on animal body weight (BW) (kg) and average daily gain (ADG)(g/day). The derived equations were highly significant (p < 0.001) and had high R2 (0.99) values. The estimated values of TDN, CP and DMI are compared with NRC (1981), Kearl (Nutrient Requirements of Ruminants in Developing Countries, All Graduate Theses and Dissertations, 1982), as well as ICAR (Livestock Management, 2013). The estimated total TDN and CP requirements at different body weights and ADG are close to the values of recommended feeding standards of Mandal et al. (Small Ruminant Res., 58, 2005, 201). The estimated DMI values are close to the values of ICAR (Livestock Management, 2013) but lower (26.5%-43.8%) as compared to NRC (1981). Regressed values are used to develop a linear programming (LP) model and a stochastic model (SM) for least-cost ration formulation for the Indian goat breed, whose average BW is about 45 kg and ADG is 130 (g/day), and which is solved using LP simplex and Generalised Reduced Gradient (GRG) nonlinear of Microsoft Excel. The models satisfy the nutrient requirement calculated by regression equations with minimum specified level of variation (usually 5%-10%) in CP and TDN. Both methods adequately meet the nutritional requirements. Therefore, an electronic sheet is developed in Excel to calculate DMI, TDN and CP for different body weights, ADG and formulate the ration by LP and stochastic model.

Social equity-based distribution networks design for the COVID-19 vaccine.

This study aims to investigate the role of social equity in vaccine distribution network design problems. Inspired by the current COVID-19 vaccine allocation in-country context, we capture social equity-based distribution by modeling three theories: Rawls' theory, Sadr's theory, and utilitarianism. We consider various social groups based on degree of urbanization, including inhabitants of cities, towns and suburbs, and rural areas. The distribution problem is subject to, on the one hand, demand-side uncertainty characterized by the daily contamination rate and its space-time propagation that anticipate the in-need population. On the other hand, supply-side uncertainty characterized by the stochastic arrival of vaccine doses for the supply period. To tackle this problem, we propose a novel bi-objective two-stage stochastic programming model using the sample average approximation (SAA) method. We also develop a lexicographic goal programming approach where the social equity objective is prioritized, thereafter reaching an efficiency level. Using publicly available data on COVID-19 in-country propagation and the case of two major provinces in Iran as example of middle-income country, we provide evidence of the benefits of considering social equity in a model-based decision-making approach. The findings suggest that the design solution produced by each social equity theory matches its essence in social science, differing considerably from the cost-based design solution. According to the general results, we can infer that each social equity theory has its own merits. Implementing Rawls' theory brings about a greater coverage percentage in rural areas, while utilitarianism results in a higher allocation of vaccine doses to social groups compared to the Sadr and Rawls theories. Finally, Sadr's theory outperforms Rawls' in terms of both the allocation and cost perspective. These insights would help decision-makers leverage the right equity approach in the COVID-19 vaccine context, and be better prepared for any pandemic crisis that the future may unfold.

A stochastic programming model for emergency supplies pre-positioning, transshipment and procurement in a regional healthcare coalition.

A regional healthcare coalition enables its member hospitals to conduct an integrated emergency supply management, which is seldom addressed in the existing literature. In this work, we propose a two-stage stochastic emergency supply planning model to facilitate cooperation and coordination in a regional healthcare coalition. Our model integrates pre-disaster emergency supplies pre-positioning and post-disaster emergency supplies transshipment and procurement and considers two planning goals, i.e., minimizing the expected total cost and the maximum supply shortage rate. With some comparison models and a case study on the West China Hospital coalition of Sichuan Province, China, under the background of the COVID-19 epidemic, we demonstrate the effectiveness and benefits of our model and obtain various managerial insights and policy suggestions for practice. We highlight the importance of conducting integrated management of emergency supplies pre-positioning, transshipment and procurement in the regional healthcare coalition for better preparation and responding to future potential disasters.

Risk-Averse Two-Stage Stochastic Minimum Cost Consensus Models with Asymmetric Adjustment Cost.

In the process of reaching consensus, it is necessary to coordinate different views to form a general group opinion. However, there are many uncertain factors in this process, which has brought different degrees of influence in group decision-making. Besides, these uncertain elements bring the risk of loss to the whole process of consensus building. Currently available models not account for these two aspects. To deal with these issues, three different modeling methods for constructing the two-stage mean-risk stochastic minimum cost consensus models (MCCMs) with asymmetric adjustment cost are investigated. Due to the complexity of the resulting models, the L-shaped algorithm is applied to achieve an optimal solution. In addition, a numerical example of a peer-to-peer online lending platform demonstrated the utility of the proposed modeling approach. To verify the result obtained by the L-shaped algorithm, it is compared with the CPLEX solver. Moreover, the comparison results show the accuracy and efficiency of the given method. Sensitivity analyses are undertaken to assess the impact of risk on results. And in the presence of asymmetric cost, the comparisons between the new proposed risk-averse MCCMs and the two-stage stochastic MCCMs and robust consensus models are also given.

Optimal surveillance against bioinvasions: a sample average approximation method applied to an agent-based spread model.

Trade-offs exist between the point of early detection and the future cost of controlling any invasive species. Finding optimal levels of early detection, with post-border active surveillance, where time, space and randomness are explicitly considered, is computationally challenging. We use a stochastic programming model to find the optimal level of surveillance and predict damages, easing the computational challenge by combining a sample average approximation (SAA) approach and parallel processing techniques. The model is applied to the case of Asian Papaya Fruit Fly (PFF), a highly destructive pest, in Queensland, Australia. To capture the non-linearity in PFF spread, we use an agent-based model (ABM), which is calibrated to a highly detailed land-use raster map (50 m × 50 m) and weather-related data, validated against a historical outbreak. The combination of SAA and ABM sets our work apart from the existing literature. Indeed, despite its increasing popularity as a powerful analytical tool, given its granularity and capability to model the system of interest adequately, the complexity of ABM limits its application in optimizing frameworks due to considerable uncertainty about solution quality. In this light, the use of SAA ensures quality in the optimal solution (with a measured optimality gap) while still being able to handle large-scale decision-making problems. With this combination, our application suggests that the optimal (economic) trap grid size for PFF in Queensland is ˜0.7 km, much smaller than the currently implemented level of 5 km. Although the current policy implies a much lower surveillance cost per year, compared with the $2.08 million under our optimal policy, the expected total cost of an outbreak is $23.92 million, much higher than the optimal policy of roughly $7.74 million.