Stochastic Integer Programming Research Articles

A stochastic optimization model for patient evacuation from health care facilities during hurricanes

We propose a rigorous modeling and methodological effort that integrates statistical implementation of hydrology models in predicting inland and coastal flood scenarios due to hurricanes and a scenario-based stochastic integer programming model which suggests resource and staging area decisions in the first stage and the evacuation decisions in the second stage. This novel study combines physics-based flood prediction models and stochastic optimization for large-scale multi-facility coordination of hospital and nursing home evacuations before impending hurricanes. The optimization model considers scenario-dependent evacuation demand, transport vehicles with varying capacities, and both critical and non-critical patients. Utilizing Hurricane Harvey of 2017 as a case study and actual healthcare facility locations in southeast Texas, we explore various evacuation policies, demonstrating the impact of routing strategies, staging area decisions, flood thresholds, and receiving facility capacities on evacuation outcomes. One of the findings is that choosing staging area(s) and deploying evacuation vehicles optimally considering the uncertainty of the hurricane’s path at the time of decision making could have significant effect on the total cost of the operation and evacuation time experienced by the evacuees. We also show the non-negligible value of the scenario-based staging and routing solution conservatively calculated in relation to a single scenario solution using the concept of value of stochastic solution.

A disaggregated integer L-shaped method for stochastic vehicle routing problems with monotonic recourse

This paper proposes a new integer L-shaped method for solving two-stage stochastic integer programs whose first-stage solutions can decompose into disjoint components, each one having a monotonic recourse function. In a minimization problem, the monotonicity property stipulates that the recourse cost of a component must always be higher or equal to that of any of its subcomponents. The method exploits new types of optimality cuts and lower bounding functionals that are valid under this property. The stochastic vehicle routing problem is particularly well suited to be solved by this approach, as its solutions can be decomposed into a set of routes. We consider the variant with stochastic demands in which the recourse policy consists of performing a return trip to the depot whenever a vehicle does not have sufficient capacity to accommodate a newly realized customer demand. This work shows that this policy can lead to a non-monotonic recourse function, but that the monotonicity holds when the customer demands are modeled by several commonly used families of probability distributions. We also present new problem-specific lower bounds on the recourse that strengthen the lower bounding functionals and significantly speed up the solving process. Computational experiments on instances from the literature show that the new approach achieves state-of-the-art results.

Integrated stochastic optimisation of stope design and long-term production scheduling at an operating underground copper mine

ABSTRACT Conventional underground long-term mine planning is based on a deterministic stepwise framework, which is unable to effectively manage the synergies between the mine planning components or to manage the orebody risk in production schedules and forecasts. The present study enhances prior integrated optimisation approaches to a variation of the sublevel longhole open stoping mining method using backfilling, through a new two-stage stochastic integer programming formulation. A comparison to a sequential stochastic approach shows different stope layouts and extraction sequences for a copper mine with secondary elements being gold and uranium. The integrated approach shows lower horizontal development costs and 6% higher net present value.

Joint stochastic optimisation of stope layout, production scheduling and access network

The three main optimisation components of sublevel stoping methods are stope layout, production schedule (or stope sequencing) and access networks. The joint optimisation of these components could further add value to an underground mining project. This potential has not been considered in the literature due to computational difficulties, and the problem was solved sequentially. This paper proposes a new joint optimisation model to integrate these components. In addition, the proposed optimisation model incorporates stochastic simulations to capture uncertainty and variability associated with the grades of the related mineral deposits mined. The optimisation model is based on a two-stage stochastic integer programming (SIP) formulation that maximises the project's net present value (NPV) and minimises the planned dilution. Applying the proposed method at a small copper deposit shows that the SIP outperforms the results obtained from mixed integer programming. For a seven-year mine life, the SIP model generated ∼20% more NPV, demonstrating the importance of developing a joint optimisation formulation and accounting for grade uncertainty and variability.

New Exact Algorithm for the integrated train timetabling and rolling stock circulation planning problem with stochastic demand

This paper studies an integrated train timetabling and rolling stock circulation planning problem with stochastic demand and flexible train composition (TRSF). A novel stochastic integer programming model, which is formulated on a space-time underlying network to simultaneously optimize the train timetable and rolling stock circulation plan with flexible train composition, is proposed by explicitly considering the random feature of passenger distribution on an urban rail transit line. To solve this problem efficiently, the proposed model is decomposed into a master problem and a series of sub-problems regarding different stochastic scenarios. We further prove that each sub-problem model is equivalent to its linear programming relaxation problem, by proving that the coefficient matrix of each linear programming relaxation model is totally unimodular. Then, the classical Benders decomposition algorithm is applied to the studied problem. Based on the model characteristics, both single-cut and multi-cut methods with some speed-up techniques are developed to solve the proposed model in a novel and effective way. Numerical experiments are conducted on small-scale cases and large-scale cases derived from Shanghai Metro Line 17, and the results show that solving the stochastic problem can extract gains in efficiency and the value of stochastic solution tends to be high.

Integrated Stochastic Underground Mine Planning with Long-Term Stockpiling: Method and Impacts of Using High-Order Sequential Simulations

The integrated optimization of stope design and underground mine production scheduling is an approach that has been shown to effectively leverage the synergies among these two underground mine planning components to generate truly optimal stope layouts and extraction sequences. The existing stochastic integrated methods, however, do not include several elements of a mining complex, such as stockpiles, due to the computational complexity and non-linearity that they might add to the optimization of the long-term mine plan. Additionally, sequential simulation methods that rely on two-point statistics and Gaussian distribution assumptions are commonly used to generate the input realizations of the mineral deposit. These methods, however, are not able to properly characterize complex spatial geometries or the high-grade connectivity of non-Gaussian and non-linear natural phenomena. The present work proposes an extension of previous developments on the integrated stope design and underground mine scheduling optimization through an expanded stochastic integer programming formulation that incorporates long-term stockpiling decisions. An application of the proposed method at an operating underground copper mine compares the cases in which the geological simulated orebody models are based on high-order and Gaussian sequential simulation methods. The extraction sequence and related final stope design are shown to be physically different. It is seen that the optimization process takes advantage of the better representation of high-grade connectivity when high-order sequential simulations are used, by targeting the areas with grades that follow the mill’s blending requirements and by making less use of the stockpiles. Overall, a 4% higher copper metal production and a resultant 6% higher net present value are observed when high-order sequential simulations are used.

Reputation-Aware Federated Learning Client Selection based on Stochastic Integer Programming

Federated Learning(FL) has attracted wide research interest due to its potential in building machine learning models while preserving users&#x0027; data privacy. However, due to the distributive nature of FL, it is vulnerable to misbehavior from participating worker nodes. Thus, it is important to select clients to participate in FL. Recent studies on FL client selection focus on the perspective of improving model training efficiency and performance, without holistically considering potential misbehavior and the cost of hiring. To bridge this gap, we propose a first-of-its-kind reputation-aware <u>S</u>tochastic integer programming-based FL <u>C</u>lient <u>S</u>election method (SCS). It can optimally select and compensate clients with different reputation profiles. Extensive experiments show that SCS achieves the most advantageous performance-cost trade-off compared to other existing state-of-the-art approaches.

Joint relocation and pricing in electric car-sharing systems

In this paper we study the integrated planning problem of determining car-sharing prices between zones of the operating area and routing employees (operators) to relocate cars in preparation for future uncertain demand. We present a novel two-stage integer stochastic programming model for this problem together with a heuristic algorithm, based on Adaptive Large Neighborhood Search (ALNS), to obtain solutions to realistically sized instances. We test the ALNS heuristic on a set of instances generated based on data from a real car-sharing organization and show that it outperforms a commercial solver.

An asynchronous parallel benders decomposition method for stochastic network design problems

Benders decomposition (BD) is one of the most popular solution algorithms for stochastic integer programs. The BD method decomposes stochastic problems into one master problem and multiple disjoint subproblems. It thus lends itself readily to parallelization. In almost all studies on the parallelization of this algorithm, the master problem remains idle until every subproblem is solved and vice versa. This can clearly result in an extremely inefficient parallel algorithm due to excessive idle times. On the other hand, relaxing the synchronization requirement can yield a nonconvergent or less efficient algorithm that may even underperform when compared to the sequential version. Addressing these issues, we introduce an asynchronous parallel BD method for stochastic network design problems. We show that the proposed algorithm converges to the global optimum and suggest various acceleration strategies to enhance its performance. We conduct an extensive numerical study on benchmark instances from a generic stochastic network design problem. The results indicate that using up to 20 processors, our asynchronous algorithm is on average 1.4 times faster than the conventional low-level parallel methods.

A study of progressive hedging for stochastic integer programming

Motivated by recent literature demonstrating the surprising effectiveness of the heuristic application of progressive hedging (PH) to stochastic mixed-integer programming (SMIP) problems, we provide theoretical support for the inclusion of integer variables, bridging the gap between theory and practice. We provide greater insight into the following observed phenomena of PH as applied to SMIP where optimal or at least feasible convergence is observed. We provide an analysis of a modified PH algorithm from a different viewpoint, drawing on the interleaving of (split) proximal-point methods (including PH), Gauss–Seidel methods, and the utilisation of variational analysis tools. Through this analysis, we show that under mild conditions, convergence to a feasible solution should be expected. In terms of convergence analysis, we provide two main contributions. First, we contribute insight into the convergence of proximal-point-like methods in the presence of integer variables via the introduction of the notion of persistent local minima. Secondly, we contribute an enhanced Gauss–Seidel convergence analysis that accommodates the variation of the objective function under mild assumptions. We provide a practical implementation of a modified PH and demonstrate its convergent behaviour with computational experiments in line with the provided analysis.

Distributed surgical scheduling across collaborating hospitals considering stochastic duration and emergency demand

In this paper, we address the need for distributed surgical scheduling across multiple hospitals and multiple days. We develop an effective ex-ante distributed schedule to determine elective service operation and elective patient assignment plan in a hospital alliance with the goal of minimizing the total expected cost considering possible realizations of treatment durations and emergency demands. The problem is formulated into a two-stage stochastic integer programming where uncertainty is characterized by a scenario set. Cancellations of scheduled elective patients and inter-hospital transfers of materialized emergency patients are explicitly modeled as second-stage decisions. To solve the problem, we propose an integer decomposition algorithm that incorporates a novel logic-based optimality cut along with a validity proof. Additionally, we introduce several enhancements to improve the algorithm. Through numerical experiments, we demonstrate that our proposed algorithm outperforms both the commercial solver and the existing integer decomposition algorithm for the proposed problem. It provides effective solutions that closely approach the best upper bounds achieved by the solver in less time, or even surpass the solver’s best upper bounds. Furthermore, we provide operational insights into other commonly-used scheduling approaches through numerical comparisons, including the strengths and limitations of the most common way of scheduling elective patients and two strategies for handling emergencies within a hospital alliance.

Efficient Algorithms for Stochastic Ride-Pooling Assignment with Mixed Fleets

Ride-pooling, which accommodates multiple passenger requests in a single trip, has the potential to substantially enhance the throughput of mobility-on-demand (MoD) systems. This paper investigates MoD systems that operate mixed fleets composed of “basic supply” and “augmented supply” vehicles. When the basic supply is insufficient to satisfy demand, augmented supply vehicles can be repositioned to serve rides at a higher operational cost. We formulate the joint vehicle repositioning and ride-pooling assignment problem as a two-stage stochastic integer program, where repositioning augmented supply vehicles precedes the realization of ride requests. Sequential ride-pooling assignments aim to maximize total utility or profit on a shareability graph: a hypergraph representing the matching compatibility between available vehicles and pending requests. Two approximation algorithms for midcapacity and high-capacity vehicles are proposed in this paper; the respective approximation ratios are [Formula: see text] and [Formula: see text], where p is the maximum vehicle capacity plus one. Our study evaluates the performance of these approximation algorithms using an MoD simulator, demonstrating that these algorithms can parallelize computations and achieve solutions with small optimality gaps (typically within 1%). These efficient algorithms pave the way for various multimodal and multiclass MoD applications. History: This paper has been accepted for the Transportation Science Special Issue on Emerging Topics in Transportation Science and Logistics. Funding: This work was supported by the National Science Foundation [Grants CCF-2006778 and FW-HTF-P 2222806], the Ford Motor Company, and the Division of Civil, Mechanical, and Manufacturing Innovation [Grants CMMI-1854684, CMMI-1904575, and CMMI-1940766]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2021.0349 .

Heterogeneous Multi-resource Planning and Allocation Under Stochastic Demand

We study the capacity planning and allocation decisions for multiple heterogeneous resources, considering potential demand scenarios, where each demand requests a subset of the available resource types simultaneously at a specified time, location, and duration (smRmD). We model this problem as a two-stage stochastic integer program and consider two variants for the objective function: (a) maximize the expected reward of demands met over all scenarios, subject to a budget B for resources, and (b) maximize the expected reward of demands met over all scenarios minus the cost of resources. Contributions of this work include (i) a thorough complexity analysis of smRmD and its variants, (ii) analysis of structural properties, (iii) development of various approximation algorithms using the unique structural properties of smRmD and its variants, and (iv) an extensive computational study to explore the ease with which exact and approximate solutions may be found. History: Accepted by Andrea Lodi, Area Editor for Design &amp; Analysis of Algorithms–Discrete. Funding: This research has been supported in part by National Science Foundation (NSF) Graduate Research Fellowship [DGE-1650044], NSF [Grants CMMI-1538860, NSF-AF:1910423, and NSF-AF:1717947], and the following Georgia Tech benefactors: William W. George, Andrea Laliberte, Richard ”Rick” E. &amp; Charlene Zalesky, and Claudia &amp; Paul Raines.

A stochastic integer programming approach to reserve staff scheduling with preferences

AbstractNowadays, reaching a high level of employee satisfaction in efficient schedules is an important and difficult task faced by companies. We tackle a new variant of the personnel scheduling problem under unknown demand by considering employee satisfaction via endogenous uncertainty depending on the combination of their preferred and received schedules. We address this problem in the context of reserve staff scheduling, an unstudied operational problem from the transit industry. To handle the challenges brought by the two uncertainty sources, regular employee and reserve employee absences, we formulate this problem as a two‐stage stochastic integer program with mixed‐integer recourse. The first‐stage decisions consist in finding the days off of the reserve employees. After the unknown regular employee absences are revealed, the second‐stage decisions are to schedule the reserve staff duties. We incorporate reserve employees' days‐off preferences into the model to examine how employee satisfaction may affect their own absence rates.

Stochastic Coded Offloading Scheme for Unmanned-Aerial-Vehicle-Assisted Edge Computing

Unmanned aerial vehicles (UAVs) have gained wide research interests due to their technological advancement and high mobility. The UAVs are equipped with increasingly advanced capabilities to run computationally intensive applications enabled by machine learning techniques. However, because of both energy and computation constraints, the UAVs face issues hovering in the sky while performing computation due to weather uncertainty. To overcome the computation constraints, the UAVs can partially or fully offload their computation tasks to the edge servers. In ordinary computation offloading operations, the UAVs can retrieve the result from the returned output. Nevertheless, if the UAVs are unable to retrieve the entire result from the edge servers, i.e., straggling edge servers, this operation will fail. In this article, we propose a coded distributed computing (CDC) approach for computation offloading to mitigate straggling edge servers. The UAVs can retrieve the returned result when the number of returned copies is greater than or equal to the recovery threshold. There is a shortfall if the returned copies are less than the recovery threshold. To minimize the cost of the network, energy consumption by the UAVs, and prevent over and under subscription of the resources, we devise a two-phase stochastic coded offloading scheme (SCOS). In the first phase, the appropriate UAVs are allocated to the charging stations amid weather uncertainty. In the second phase, we use the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$z$ </tex-math></inline-formula> -stage stochastic integer programming (SIP) to optimize the number of computation subtasks offloaded and computed locally, while taking into account the computation shortfall and demand uncertainty. By using a real data set, the simulation results show that our proposed scheme is fully dynamic and minimizes the cost of the network and UAV energy consumption amid stochastic uncertainties.

Risk-based pavement maintenance planning considering budget and pavement deterioration uncertainty

Although some parameters are uncertain in pavement Maintenance and Rehabilitation (M&R) planning, most of the literature has considered these parameters deterministic. Likewise, minimizing risk has been an immense concern as a significant amount of money should be spent on the M&R of pavement networks. To this end, this study aims to propose a new approach to consider risk and uncertainty in a pavement M&R optimization problem. A Multistage Stochastic Integer Programming including risk constraints is applied for the problem modeling. Conditional Value at Risk index is applied to minimize the mentioned variables and avoid unacceptable risks. Subsequently, a case study is used to analyze the proposed technique’s effectiveness. The results indicate that the total pavement M&R agency cost equals 6.33 billion Tomans considering a confidence level of 95%. Furthermore, the proposed approach can detect the optimal value of the annual budget for pavement network M&R, which is 1.96 billion Tomans for the case study. Applying preventive maintenance can compensate for the negative effects of risk on projects. Therefore, the proposed approach can help decision-makers, policymakers, and transport agencies to analyze and control risk in the pavement M&R optimization problems and determine the optimal M&R budget.

RBDMS: Rate-Adaptation and Buffer-Awareness Data Gathering for Mobile Sink Scheduling in WSNs

Employing a mobile sink (MS) to act as a relay node in wireless sensor network (WSN) applications is a promising solution for efficient power saving and data collection. However, establishing long-distance traveling leads to larger latency or inefficient buffer management at rendezvous points (RPs), e.g., flying UAVs in disaster management. Moreover, there is no efficient solution to guarantee the completeness of data gathering by considering the waiting time (sojourn) of the MS to receive packets from RPs in addition to the MS moving time among RPs. This work presents a rate-adaptation and buffer-awareness data gathering for MS scheduling (RBDMS) by constructing grid cells in the monitoring area. In fact, it establishes the shortest path by passing within the communication range of the sensors based on data volume. RBDMS not only has a mechanism for handling emergency packets with low latency but also benefits from stochastic integer programming (SIP) for scheduling the MS sojourn time with lower computational time using Lagrangian relaxation. Simulation results confirm that the proposed RBDMS outperforms comparable state-of-the-art works in terms of the MS path length, network lifetime, the energy consumption of sensors, and MS, as well as buffering performance.

Stochastic service network design for a platooning service provider

Automated driving is expected to support a wide range of transportation applications in the near future. However, especially in urban environments, not all streets may be feasible for automated driving which restricts the operation of autonomous vehicles (AVs) to certain zones. Outside of these zones, AVs may be safely navigated by a leading vehicle that is operated by a human driver using the concept of platooning. In this paper, we consider the service network design problem of a service provider that offers platooning services to AV operators in an urban environment in exchange for monetary compensation. Tactical planning decisions comprise determining the fleet size and scheduling platooning services for a planning horizon such as a season. Those decisions are supported by the quoting of rates to potential customers that reflect their willingness to pay depending on the received service level. A second stage of decisions, made on each day of the planning horizon, comprises the acceptance or rejection of individual platooning requests and the routing of AVs through the platooning services. Since the platooning demand is assumed to be stochastic, we model the problem as a two-stage stochastic integer program. On a real-world based network, we perform computational experiments that indicate potential profits for a platooning service provider and considerable savings for AV operators with different demand characteristics. We further assess the value of considering uncertainty.

Вычислительный эксперимент в оптимизационном моделировании процесса мультимодальных грузоперевозок на основе эгалитарных принципов

Purpose: To build an optimization model of multimodal cargo transportations in a port-side transport and technological system with the participation of rail and sea kinds of transport based on egalitarian principles, the model allows manipulating the values of parameters characterizing a transportation process and taking into account interests of transportation process participants within the frames of being considered cost indicators. Methods: An algorithm has been developed for planning the shipment and delivery of cargo flows to port station address in view of routing, time and cost parameters of transportation process which under, cost optimal values of being transported cargo and complex time indicators of transportation process are provided. One of the limitations in the optimization task is the volume of a ship batch that’s expected at any port unloading stations. The built mathematical model is a multi-criteria and multi-extreme task of nonlinear stochastic integer programming. The research theoretical and economic basis is the synthesis of the classical concepts of utilitarian and egalitarian approaches in a welfare theory that makes it possible to build sounded theoretically and demanded in practice the optimization models of multimodal freight transportation railway component. Results: Developed in analytical computing system environment the optimization algorithm provides opportunities along with Pareto optimal transportation plans for to find plans embodying “just in time” logistic concept, so at these plans, it is also achieved the balance of interests of transportation considered participants. Practical importance: An approach is proposed for to study freight transportation organization on a railway landfill wherein transportations are pursued within the framework of multimodal port-side transport and technological system. The developed models and algorithms can be recommended for practical use during the formation and analysis of logistic supply chains with rail and sea transport kind participation.

A stochastic dual dynamic integer programming based approach for remanufacturing planning under uncertainty

ABSTRACT We seek to optimize the production planning of a three-echelon remanufacturing system under uncertain input data. We consider a multi-stage stochastic integer programming approach and use scenario trees to represent the uncertain information structure. We introduce a new dynamic programming formulation that relies on a partial nested decomposition of the scenario tree. We then propose a new approximate stochastic dual dynamic integer programming algorithm based on this partial decomposition. Our numerical results show that the proposed solution approach is able to provide near-optimal solutions for large-size instances with a reasonable computational effort.