Stochastic Mixed-integer Programming Research Articles

Project Selection and Scheduling with Multiplicative Enhancement Effects and Delay Risk: An Application in Intelligent Manufacturing Technologies

In the realm of intelligent manufacturing, driven by the push towards smart factories and Industry 4.0, optimizing technology selection and sequencing is paramount for intelligent transformation. This complex decision-making process resembles a novel project selection and scheduling problem, characterized by multiplicative enhancement effects (MEEs) where the collective benefits of multiple projects can exceed their individual contributions. Additionally, the uncertain duration of projects further adds complexity. Motivated by these practical challenges, a stochastic mixed-integer programming model that incorporates MEEs and uncertain delays is established. A conditional value-at-risk (CVaR)-based risk measure is integrated to control delay risk. The solution approach leverages an efficient branch and bound (B&B)-based approach with efficient cutting strategies, integrating a sample average approximation (SAA) framework and a backward labeling algorithm to handle stochastic elements. A real-world technology implementation case study highlights the advantages of considering MEEs, uncovers myopic biases in technology selection, and identifies implementation patterns centred around core Industry 4.0 technologies. This research enhances our understanding of intelligent manufacturing decision-making, offering valuable insights and practical implications for technology-driven transformations.

A stochastic programming approach to the antibiotics time machine problem

Antibiotics Time Machine is an important problem to understand antibiotic resistance and how it can be reversed. Mathematically, it can be modeled as follows: Consider a set of genotypes, each of which contain a set of mutated and unmutated genes. Suppose that a set of growth rate measurements of each genotype under a set of antibiotics is given. The transition probabilities of a ‘realization’ of a Markov chain associated with each arc under each antibiotic are computable via a predefined function given the growth rate realizations. The aim is to maximize the expected probability of reaching to the genotype with all unmutated genes given the initial genotype in a predetermined number of transitions, considering the following two sources of uncertainties: (i) the randomness in growth rates, (ii) the randomness in transition probabilities, which are functions of growth rates. We develop stochastic mixed-integer linear programming and dynamic programming approaches to solve static and dynamic versions of the Antibiotics Time Machine Problem under the aforementioned uncertainties. We adapt a Sample Average Approximation approach that exploits the special structure of the problem and provide accurate solutions that perform very well in an out-of-sample analysis.

Resource allocation and route planning under the collection price uncertainty for the biomass supply chain

The escalating severity of global warming has drawn worldwide attention to ecological problems. Nevertheless, with the introduction of environmental policies, biomass resources have been effectively developed as a renewable energy source. This paper investigates an advanced biomass supply chain design wherein biomass resources are initially converted into bio-oil through widely distributed fast pyrolysis facilities, and are subsequently transported to a centralised biorefinery for further refining into biofuels. This novel biomass supply chain addressed three key issues: (1) The number of fast pyrolysis facilities, (2) The allocation of resources, and (3) The routes of resources transport. In respect of these problems, a two-stage stochastic mixed integer programming model is established to minimise the total cost of the biomass supply chain considering the uncertainty collection price of fast pyrolysis facilities. A hybrid simulated annealing algorithm which incorporates the sample average approximation method is proposed to solve the stochastic model and is effectiveness for large-scale examples. Finally, a sensitivity analysis is performed using the proposed algorithm and the results show that the proposed stochastic model outperforms the deterministic model under uncertain collection price. The model allows optimising the biomass supply chain economic performances and minimise financial risk on investment by determining the fast pyrolysis facility locations, reasonable resource allocation and optimal transport routes under uncertain collection price.

Disassembly line balancing with hazardous task failures – Model based solution approaches

In this study, we consider a disassembly line balancing problem that involves a fixed number of workstations and the possibility of failures in hazardous tasks. We assume that each hazardous task may fail with a pre-specified probability and once it fails, the disassembly line stops, and all tasks assigned to subsequent workstations are interrupted. Our problem is to select the set of tasks for processing and assign them to the workstations to maximize the total expected net revenue.We develop a heuristic procedure that uses the linear programming relaxation of the recently reported stochastic mixed-integer program in the literature. To assess the performance of the heuristic procedure, we use two upper bounds that use some properties of the optimal solutions as valid cuts.Our extensive experiments have shown the excellent performance of our heuristic algorithm and upper bounds.

An Integrated Predictive Maintenance and Operations Scheduling Framework for Power Systems Under Failure Uncertainty

Maintenance planning plays a key role in power system operations under uncertainty as it helps providers and operators ensure a reliable and secure power grid. This paper studies a short-term condition-based integrated maintenance planning with operations scheduling problem while considering the possible unexpected failure of generators as well as transmission lines. We formulate this problem as a two-stage stochastic mixed-integer program with failure scenarios sampled from the sensor-driven remaining lifetime distributions of the individual system elements whereas a joint chance constraint consisting of Poisson Binomial random variables is introduced to account for failure risks. Because of its intractability, we develop a cutting-plane method to obtain an exact reformulation of the joint chance constraint by proposing a separation subroutine and deriving stronger cuts as part of this procedure. We also derive a second-order cone programming-based safe approximation of this constraint to solve large-scale instances. Furthermore, we propose a decomposition algorithm implemented in parallel fashion for solving the resulting stochastic program, which exploits the features of the integer L-shaped method and the special structure of the maintenance and operations scheduling problem to derive valid and stronger sets of optimality cuts. We further present preprocessing steps over transmission line flow constraints to identify redundancies. To illustrate the computational performance and efficiency of our algorithm compared with more conventional maintenance approaches, we design a computational study focusing on a weekly plan with daily maintenance and hourly operational decisions involving detailed unit commitment subproblems. Our computational results on various IEEE instances demonstrate the computational efficiency of the proposed approach with reliable and cost-effective maintenance and operational schedules. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2022.0154 .

Model and solution approach to coordinate production-inventory strategies considering nonlinear price-sensitive demand: application to Canadian pulp and paper industry

This study addresses a practical problem within a multi-level supply chain where a wide range of customers can be served through different strategies such as make-to-stock, make-to-order, or vendor-managed inventory. The customer demand is stochastic, and sensitive to pricing associated with different production-inventory strategies. We propose a two-stage stochastic mixed-integer non-linear programming model. In the first stage, decisions are made regarding the selection of production-inventory strategies and pricing to maximise the expected profit. The second stage involves decisions related to production, inventory, and distribution, which are used to evaluate the first-stage decisions under various scenarios with different levels of accuracy. To solve the model, a metaheuristic approach based on the Simulated Annealing algorithm is developed. To showcase the practical applicability of our model and solution approach, we use a real case study in a Canadian pulp and paper supply chain. The results revealed that both the production-inventory strategy assigned to customers and the sales price underwent changes across scenarios. Furthermore, we demonstrated that by implementing the SA algorithm, we could improve the initial profit by up to 1.43% through slight adjustments in the sales price and assigned strategies for customers.

Heuristic approach for optimising reliable supply chain network using drones in last-mile delivery under uncertainty

ABSTRACT In addressing the challenges of last-mile logistics, the reliability of the supply chain network becomes paramount. These challenges are intensified due to drone performance limitations and various uncertainties in supply chain operations. While recent literature recognises the potential of drones for last-mile delivery, it falls short in effectively considering these uncertainties in drone-enabled supply chain models. Our research addresses this gap with two major contributions: first, a novel stochastic mixed-integer programming model is developed to construct a feasible delivery network, including warehouses and recharging stations, enhancing both coverage and reliability. Second, a modification in the genetic algorithm by considering each scenario independently improves computational efficiency, outperforming commercial software by an average of 40% and up to 55%. Empirical findings reveal that strategic investments in system hardening can yield substantial improvements in reliability. Despite the absence of real-world stochastic parameters as a limitation, this research pioneers the design of reliable networks under uncertainties and extends drone coverage through strategic charging stations. This work sets a significant milestone for future optimization in drone logistics, offering practical implications for supply chain managers considering drone adoption.

Unified Branch-and-Benders-Cut for two-stage stochastic mixed-integer programs

Two-stage stochastic programs are a class of stochastic problems where uncertainty is discretized into scenarios, making them amenable to solution approaches such as Benders decomposition. However, classic Benders decomposition is not applicable to general two-stage stochastic mixed-integer programs due to the restriction that the second-stage variables should be continuous. We propose a novel Benders decomposition-based framework that accommodates mixed-integer variables in both stages as well as uncertainty in all of the recourse parameters. The proposed approach is a unified branch-and-Benders algorithm, where we use a heuristic to maintain a global upper bound and a post-processing phase to determine an optimal solution. This new approach is flexible, allowing practitioners to integrate acceleration techniques such as partial decomposition or convexification schemes. We demonstrate the efficiency of our approach versus classic ones on the stochastic server location problem, and its generality on a new, complex stochastic problem where the second stage is a traveling salesman problem.

A novel cost-based optimization model for electric power distribution systems resilience improvement under dust storms

In the recent years, dust storms (DSs) pose a serious threat to critical infrastructure such as power distribution networks (PDNs). During DSs, the contamination of insulators, increases the possibility of damage to the PDNs insulation system and flashover induced power outage may occur. Power outages disrupt the performance of other urban infrastructures and, in addition to heavy financial losses, cause public dissatisfaction. Although this issue is of particular importance in areas with humid climate, a few studies have been reported on PDNs resilience improvement against DSs. This paper proposes a novel cost-based optimization model to make PDNs more resilient to DSs considering uncertainties. The proposed model is based on the two-stage stochastic mixed-integer programming (SMIP). In the first stage, decisions are made to equip repair crews (RCs) with insulator washing machines, hardening distribution lines with silicone-rubber insulators (SIs), and deploy backup distributed generators (DGs). Decisions in the second stage include network reconfiguration, RCs routing, DGs power dispatch, and load shedding as the critical options for PDN outage management during/after DSs. Case studies are evaluated in the IEEE 69-bus test system and a real 209-bus PDN in Khuzestan province, a coastal province in southwestern Iran. The simulation results at different budget levels have confirmed the efficiency of the proposed model for cost-optimal resilience enhancement planning of PDNs against DSs.

Ensemble Variance Reduction Methods for Stochastic Mixed-Integer Programming and their Application to the Stochastic Facility Location Problem

Sample average approximation (SAA), the standard approach to stochastic mixed-integer programming, does not provide guidance for cases with limited computational budgets. In such settings, variance reduction is critical in identifying good decisions. This paper explores two closely related ensemble methods to determine effective decisions with a probabilistic guarantee. (a) The first approach recommends a decision by coordinating aggregation in the space of decisions as well as aggregation of objective values. This combination of aggregation methods generalizes the bagging method and the “compromise decision” of stochastic linear programming. Combining these concepts, we propose a stopping rule that provides an upper bound on the probability of early termination. (b) The second approach applies efficient computational budget allocation for objective function evaluation and contributes to identifying the best solution with a predicted lower bound on the probability of correct selection. It also reduces the variance of the upper bound estimate at optimality. Furthermore, it adaptively selects the evaluation sample size. Both approaches provide approximately optimal solutions even in cases with a huge number of scenarios, especially when scenarios are generated by using oracles/simulators. Finally, we demonstrate the effectiveness of these methods via extensive computational results for “megascale” (extremely large scale) stochastic facility location problems. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Funding: This work was supported by The Office of Naval Research [Grant N00014-20-1-2077] and the Air Force Office of Scientific Research [Grant FA9550-20-1-0006]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2021.0324 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2021.0324 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Toward equitable grid resilience: operationalizing climate adaptation strategies to mitigate flooding impacts

Disadvantaged communities are disproportionately affected by flooding, exacerbated by climate change. This paper presents a novel framework for incorporating environmental justice into climate adaptation planning of power grids against flooding. A new energy equity metric is introduced with the vision that addressing environmental justice warrants prioritizing disadvantaged communities that have lower risk thresholds. The framework is applied to a levee-protected IEEE standard test system in northern California. The grid performance disturbed due to flooded substations is investigated under current and future climate. The mathematical model of the framework is structured as a two-stage stochastic mixed-integer programming model. This model aims to minimize the equity gap in grid resilience (EGGR) between disadvantaged and non-disadvantaged communities while enhancing the system resilience by reducing the risk of power outages due to flooding. The results show that climate change undermines grid resilience, with disproportionally worse impacts on disadvantaged communities. A significant EGGR is observed that worsens under a changing climate. For adaptation, the optimal placement of distributed energy resources is determined by maximizing the grid resilience to flooding while minimizing EGGR. The proposed framework can equip decision-makers with a robust tool for operationalizing equitable climate adaptation strategies for power grids.

Multiserver time window allowance schedules for virtual visits with uncertain time-dependent no-shows and service times

Virtual care serves as a new mode that can divert non-urgent visits from traditional office visits. Whether virtual service can improve the access to medical treatment and reduce the burden of traditional office services, the key issue is to generate efficient appointment schedules with the lowest operation cost. In this paper, considering the uncertainty of time-dependent no-shows and service times, we investigate a multiserver time window allowance appointment scheduling problem, where time window constraints that restricts virtual visits to be served during the particular period are explicitly modeled. We formulate the problem as a stochastic mixed-integer program to optimize decisions of physician allocation and appointment time simultaneously. Based on the sample average approximation, a stabilized Benders decomposition algorithm is developed by incorporating acceleration techniques, such as cut aggregation and feasibility cuts. Numerical results based on real data indicate the effectiveness of the proposed multiserver time window allowance schedules (MTWAS) and algorithm. Comparing with the off-the-shelf solver Gurobi, the developed algorithm demonstrates high performance in terms of computation speed and solution quality. Under different time-dependent no-show patterns of virtual and office visits, the obtained MTWAS perform better than previous solutions in almost all test cases. In addition, we offer useful managerial insights to aid the virtual service provider in making better scheduling decisions.

Integrated optimization for high-speed railway express system with multiple modes

High-speed railway (HSR) express, as an emerging transport option in the logistics industry, provides sufficient transport resources to satisfy the demands for large-scale, high value-added, and customized logistics requirements. Unlike conventional transport options, the HSR express is capable of using multiple modes (i.e., piggybacking, reserved-coach, and freight trains modes) to efficiently convey freights to a broad variety of locations. Based on the operational characteristics of various HSR express modes, this study takes into consideration of integrated optimization problems, such as whether high-speed trains of various HSR express modes can transport the freights of specific types, the amounts of freights transported by each HSR express mode, and the arrangement of capacity resources of each mode. This study establishes a stochastic mixed integer programming model with the goal of maximizing the net profit of the HSR express system and designs a heuristic solution approach to solve the model efficiently. To verify the validity of the proposed model and algorithm, a large number of numerical experiments and a real-world case are conducted. Based on the extensive experiments, this study provides railway companies potentially with useful insights for developing the HSR express with various modes.

Data-driven joint optimization of maintenance and spare parts provisioning for deep space habitats: A stochastic programming approach

The utilization of sensor data to make informed decisions on maintenance schedule and spare parts provisioning will be determinant to the development of self-aware habitable facilities–so called SmartHabs–in deep space regions. To address this novel and challenging problem, this work presents a chance constrained stochastic mixed integer program to jointly optimize maintenance and spare parts provisioning for ensuring reliable and cost-efficient operations of SmartHabs. A SmartHab is modeled as a multi-unit system with non-identical components that randomly degrade over time. The proposed optimization model is driven by telemetry sensor data as the uncertainty of the remaining lifetime of the components is estimated by data-driven prognostic algorithms. The model further incorporates characteristic operational conditions of deep space habitats such as a long planning horizon that requires multiple maintenance decisions for each component, maintenance task assignment to humans and robots, long lead times between resupply trips, and high penalties due to stock-outs of spare parts. The formulation introduces chance constraints restricting the unavailability of critical components and the number of corrective repairs with high probability under the uncertainty of component failures. The nonlinearities induced by the chance constraints are circumvented by the derivation of two linearization methods based on scenario representation and safe approximations, resulting in their tractable mixed integer linear reformulations. The proposed model is validated by simulations, showing reliable and effective plans against benchmark approaches while evidencing to be computationally scalable as the number of components increases.

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.

Buffer Times Between Scheduled Events in Resource Assignment Problem: A Conflict-Robust Perspective

Problem definition: In many resource scheduling problems for services with scheduled starting and completion times (e.g., airport gate assignment), a common approach is to maintain appropriate buffer between successive services assigned to a common resource. With a large buffer, the chances of a “crossing” (i.e., a flight arriving later than the succeeding one at the gate) will be significantly reduced. This approach is often preferred over more sophisticated stochastic mixed-integer programming methods that track the arrival of all the flights to infer the number of “conflicts” (i.e., a flight arriving at a time when the assigned gate becomes unavailable). We provide a theoretical explanation, from the perspective of robust optimization for the good performance of the buffering approach in minimizing not only the number of crossings but also the number of conflicts in the operations. Methodology/results: We show that the buffering method inherently minimizes the worst-case number of “conflicts” under both robust and distributionally robust optimization models using down-monotone uncertainty sets. Interestingly, under down-monotone properties, the worst-case number of crossings is identical to the worst-case number of conflicts. Using this equivalence, we demonstrate how feature information from flight and historical delay information can be used to enhance the effectiveness of the buffering method. Managerial implications: The paper provides the first theoretical justification on the use of buffering method to control for the number of conflicts in resource assignment problem. Funding: This work was supported by the 2019 Academic Research Fund Tier 3 of the Ministry of Education-Singapore [Grant MOE-2019-T3-1-010] and the Research Grants Council of Hong Kong SAR, China [Grant PolyU 152240/17E]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2022.0572 .

Designing a stochastic supply chain network: An error-bound-based heuristic

Designing a stochastic supply chain is challenging because demand is often partially observable or unknown in advance. This paper presents an alternative approach to address a three-echelon stochastic supply chain network design with insufficient demand information. We formulate a stochastic mixed-integer program with chance constraints. We identify a lower and upper bound on the value at risk to deal with chance constraints, thus inducing two mixed-integer linear approximations to the stochastic program. We develop a branch-and-price scheme to solve the approximations and a primal heuristic algorithm for initial columns. We then create a heuristic algorithm for solving the original problem by implementing sensitivity analysis and error bound on the approximations. Besides network size, computational experiments find that network design performance depends on aggregate service levels and maximum demand variations. The aggregate behavior of service levels has an inverse impact on the network design performance, while maximum coefficients of demand variations present a positive effect. Computational results reveal that our proposed algorithms perform well compared to normal distributions and a robust optimization formulation. Our models and algorithms suggest a feasible network design tool for handling uncertainties in supply chains.

The Multi-Trip Autonomous Mobile Robot Scheduling Problem with Time Windows in a Stochastic Environment at Smart Hospitals

Autonomous mobile robots (AMRs) play a crucial role in transportation and service tasks at hospitals, contributing to enhanced efficiency and meeting medical demands. This paper investigates the optimization problem of scheduling strategies for AMRs at smart hospitals, where the service and travel times of AMRs are stochastic. A stochastic mixed-integer programming model is formulated to minimize the total cost of the hospital by reducing the number of AMRs and travel distance while satisfying constraints such as AMR battery state of charge, AMR capacity, and time windows for medical requests. To address this objective, some properties of the solutions with time window constraints are identified. The variable neighborhood search (VNS) algorithm is adjusted by incorporating the properties of the AMR scheduling problem to solve the model. Experimental results demonstrate that VNS generates high-quality solutions. Both enhanced efficiency and the meeting of medical demands are achieved through intelligently arranging the driving routes of AMRs for both charging and service requests, resulting in substantial cost reductions for hospitals and enhanced utilization of medical resources.

Reinforcement learning and mixed-integer programming for power plant scheduling in low carbon systems: Comparison and hybridisation

Decarbonisation is driving dramatic growth in renewable power generation. This increases uncertainty in the load to be served by power plants and makes their efficient scheduling, known as the unit commitment (UC) problem, more difficult. UC is solved in practice by mixed-integer programming (MIP) methods; however, there is growing interest in emerging data-driven methods including reinforcement learning (RL). In this paper, we extensively test two MIP (deterministic and stochastic) and two RL (model-free and with lookahead) scheduling methods over a large set of test days and problem sizes, for the first time comparing the state-of-the-art of these two approaches on a level playing field. We find that deterministic and stochastic MIP consistently produce lower-cost UC schedules than RL, exhibiting better reliability and scalability with problem size. Average operating costs of RL are more than 2 times larger than stochastic MIP for a 50-generator test case, while the cost is 13 times larger in the worst instance. However, the key strength of RL is the ability to produce solutions practically instantly, irrespective of problem size. We leverage this advantage to produce various initial solutions for warm starting concurrent stochastic MIP solves. By producing several near-optimal solutions simultaneously and then evaluating them using Monte Carlo methods, the differences between the true cost function and the discrete approximation required to formulate the MIP are exploited. The resulting hybrid technique outperforms both the RL and MIP methods individually, reducing total operating costs by 0.3% on average.

Designing a resilient and green coal supply chain network under facility disruption and demand volatility

Few studies have addressed strategic-level disruptions and operational-level risks for the coal industry in the literature on mining supply chain management. In many developing countries, the coal industry plays a leading role in electric power generation and primarily contributes to carbon emissions. This study aims to design a resilient and green coal supply chain network (RG-CSCN) considering strategic-level disruptions, operational-level risks, and carbon emissions. A practical stochastic mixed-integer programming model is developed to optimise both economic and environmental objectives (i.e., the total network cost and the total carbon emissions cost). Our proposed model involves five resilience strategies: facility fortification, reserving emergency inventory, direct-to-port delivery, establishing reliable distribution centres, and multiple transportation routes. Based on real-world data from the coal industry, we conduct extensive computational experiments to evaluate the impacts of strategic-level disruptions and operational-level risks and to validate the applicability of the proposed five resilience strategies and their synergistic performance in coal supply chain management. Moreover, sensitivity analyses are carried out to examine the influences of different customer service levels and carbon tax policies in various scenarios.