Stochastic Mixed-integer Programming Research Articles

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.

A Converging Benders’ Decomposition Algorithm for Two-Stage Mixed-Integer Recourse Models

Novel Optimality Cuts for Two-Stage Stochastic Mixed-Integer Programs The applicability and use of two-stage stochastic mixed-integer programs is well-established, thus calling for efficient decomposition algorithms to solve them. Such algorithms typically rely on optimality cuts to approximate the expected second stage cost function from below. In “A Converging Benders’ Decomposition Algorithm for Mixed-Integer Recourse Models,” van der Laan and Romeijnders derive a new family of optimality cuts that is sufficiently rich to identify the optimal solution of two-stage stochastic mixed-integer programs in general. That is, general mixed-integer decision variables are allowed in both stages, and all data elements are allowed to be random. Moreover, these new optimality cuts require computations that decompose by scenario, and thus, they can be computed efficiently. Van der Laan and Romeijnders demonstrate the potential of their approach on a range of problem instances, including the DCAP instances from SIPLIB.

Transmission Defense Hardening Against Typhoon Disasters Under Decision-Dependent Uncertainty

Transmission defense hardening (TDH) is an effective practice to ensure the safe operation of the power system during and after natural disasters, such as typhoons. However, the coupling among the typhoon disaster, hardening decision, and transmission line status is not addressed in the previous research works. Besides, the enhancement measures for different types of transmission line failures have not yet been assessed. This paper proposes a comprehensive framework to improve power transmission system resilience against typhoon disasters. Firstly, the motion path and the wind field of typhoons are simulated using Monte Carlo sampling to quantify the spatiotemporal impacts of wind speed on the transmission line status. The decision-dependent uncertainty (DDU), which reflects the effects of hardening decisions on the transmission lines, is addressed by sampling according to the corresponding failure probabilities. Then, the scenario simulation is integrated into the proposed two-stage stochastic mixed-integer programming (SMIP) model considering various enhancement measures for different failures. Since the way to handle the DDU needs a large number of scenarios, it leads to high computation complexity. Hence, the sample average approximation (SAA) algorithm is introduced to cope with it. Finally, the proposed framework and its solution algorithm are carried out on the modified IEEE RTS-79 system and the modified IEEE 118-Bus system. The significant cost saving and resilience improvement demonstrate the effectiveness of the proposed framework.

Pre-disaster and mid-disaster resilience improvement strategy of high proportion renewable energy system considering frequency stability

This paper aims to investigate the influence of dynamic system frequency changes on the resilience improvement process of high proportion renewable energy systems (HPRES). To this end, a module mobile energy storage system (MMESS) is considered to participate in primary frequency modulation and two types of system frequency stability constraints are established to guarantee the safety of frequency nadir and maximum frequency drop rate. Furthermore, taking into account the uncertainty of fault components caused by extreme weather, a two-stage stochastic mixed-integer programming (SMIP) is proposed for pre-disaster preventive control. By using progressive hedging method, generator and MMESSs scheduling plans can be converged after multiple iterations. For a deterministic fault scenario, a single-stage mixed-integer linear programming is adopted for mid-disaster emergency dispatch. Based on results from pre-disaster process, generators and MMESSs are coordinated to optimize load reduction while ensuring system frequency stability. Finally, case studies are conducted to demonstrate that the proposed strategy can effectively improve HPRES resilience while reducing renewable energy threat to frequency stability.

Vehicle Rebalancing in a Shared Micromobility System with Rider Crowdsourcing

Problem definition: Shared micromobility vehicles provide an eco-friendly form of short-distance travel within an urban area. Because customers pick up and drop off vehicles in any service region at any time, such convenience often leads to a severe imbalance between vehicle supply and demand in different service regions. To overcome this, a micromobility operator can crowdsource individual riders with reward incentives in addition to engaging a third-party logistics provider (3PL) to relocate the vehicles. Methodology/results: We construct a time-space network with multiple service regions and formulate a two-stage stochastic mixed-integer program considering uncertain customer demands. In the first stage, the operator decides the initial vehicle allocation for the regions, whereas in the second stage, the operator determines subsequent vehicle relocation across the regions over an operational horizon. We develop an efficient solution approach that incorporates scenario-based and time-based decomposition techniques. Our approach outperforms a commercial solver in solution quality and computational time for solving large-scale problem instances based on real data. Managerial implications: The budgets for acquiring vehicles and for rider crowdsourcing significantly impact the vehicle initial allocation and subsequent relocation. Introducing rider crowdsourcing in addition to the 3PL can significantly increase profit, reduce demand loss, and improve the vehicle utilization rate of the system without affecting any existing commitment with the 3PL. The 3PL is more efficient for mass relocation than rider crowdsourcing, whereas the latter is more efficient in handling sporadic relocation needs. To serve a region, the 3PL often relocates vehicles in batches from faraway, low-demand regions around peak hours of a day, whereas rider crowdsourcing relocates a few vehicles each time from neighboring regions throughout the day. Furthermore, rider crowdsourcing relocates more vehicles under a unimodal customer arrival pattern than a bimodal pattern, whereas the reverse holds for the 3PL. Funding: This work was supported by the Research Grants Council of Hong Kong [Grants 15501319 and 15505318] and the National Natural Science Foundation of China [Grant 71931009]. Z. Jin was supported by the Hong Kong PhD Fellowship Scheme. Y. F. Lim was supported by the Lee Kong Chian School of Business, Singapore Management University [Maritime and Port Authority Research Fellowship]. Supplemental Material: The online appendices are available at https://doi.org/10.1287/msom.2023.1199 .

Reshore or not Reshore: A Stochastic Programming Approach to Supply Chain Optimization

This paper presents a scenario-based stochastic mixed integer programming model for risk-neutral or risk-averse optimization of supply chain reshoring to domestic region, under the ripple effect propagated from a foreign disruption source region. The reshoring decisions with respect to tier-one suppliers of parts and tier-zero OEM (Original Equipment Manufacturer) assembly plants are considered under disruptions in supply, manufacturing, logistics and demand rippling across the entire supply chain. The proposed innovative approach integrates strategic supply chain reshoring and operational supply chain scheduling, which allows the decision maker to evaluate the operational impact of the strategic decision. Results of computational experiments, partially modeled after a supply chain reshoring problem in the smartphone industry, are provided. The findings indicate that reshoring decisions are strongly dependent on the level of government subsidy for capital expenditure and for risk-neutral reshoring, a portfolio of supply chain nodes with positive expected net savings can be considered only. In general, the reshored supply chain can better meet domestic market demand. Moreover, full reshoring of a supply chain improves its business as usual performance and even partial reshoring mitigates the impact of the ripple effect. However, for risk-averse decision-making, if reshoring is incapable of reducing worst-case cost, in particular, worst-case lost sales, no reshoring is selected.

Routing and resource allocation in non-profit settings with equity and efficiency measures under demand uncertainty

Motivated by food distribution operations for non-profit organizations, we study a variant of the stochastic routing-allocation problem under demand uncertainty, in which one decides the assignment of trucks for demand nodes, the sequence of demand nodes to visit (i.e., truck route), and the allocation of food supply to each demand node. We propose three stochastic mixed-integer programming (SMIP) models representing different performance measures important to food banks, namely maximizing efficiency, maximizing equity, and maximizing efficiency and equity simultaneously. To solve practical large-scale instances, we develop an original matheuristic based on adaptive large-scale neighborhood search. Using real-world data based on real-life instances, we conduct an extensive numerical experiment to assess the computational performance of our approach and derive insights relevant to food banks. The proposed matheuristic produces high-quality solutions quickly with an optimality gap never exceeding 4.11% on tested instances. We also demonstrate the performance of the three models in terms of service levels, food waste, and equity.

Integer set reduction for stochastic mixed-integer programming

Two-stage stochastic mixed-integer programs (SMIPs) with general integer variables in the second-stage are generally difficult to solve. This paper develops the theory of integer set reduction for characterizing a subset of the convex hull of feasible integer points of the second-stage subproblem which can be used for solving the SMIP with pure integer recourse. The basic idea is to use the smallest possible subset of the subproblem feasible integer set to generate a valid inequality like Fenchel decomposition cuts with a goal of reducing computation time. An algorithm for obtaining such a subset based on the solution of the subproblem linear programming relaxation is devised and incorporated into a decomposition method for SMIP. To demonstrate the performance of the new integer set reduction methodology, a computational study based on randomly generated knapsack test instances was performed. The results of the study show that integer set reduction aids in speeding up cut generation, leading to better bounds in solving SMIPs with pure integer recourse than using a direct solver.

Optimal mining cut definition and short-term open pit production scheduling under geological uncertainty

Short-term planners must define an operational schedule based on blasthole data, which might be incomplete or imperfect. As a result, there is uncertainty in the true grade of each Selective Mining Unit. This can impact the definition of mining cuts and dig-limits, and the fulfillment of long-term targets. In this work, we propose a stochastic mixed integer program to define an operational mining cut configuration and a short-term schedule under grade uncertainty. The objective of the model is to maximize revenue and minimize deviations from production targets. We test the model using real data from a copper deposit and use grade simulations for quantifying the uncertainty in blasthole data. The results show that the stochastic model achieves higher profit margins and lower deviations from production targets compared to a deterministic variant based on a single estimated scenario. At the same time, the numerical experiments show the potential impact of blasthole uncertainty in profit and compliance of short-term schedules. Therefore, the use of this stochastic formulation can help planners find optimal mining cuts and improve target fulfillment and profitability of short-term operational plans in the presence of grade uncertainty. Finally, we propose several extensions to include new sources of uncertainty.

Multi-mitigation strategies in medical supplies for epidemic outbreaks

The outbreak of Coronavirus disease 2019 (COVID-19) highlights the importance of sufficient medical supplies stockpiling at the pre-event stage. In contrast, the potential disadvantages of maintaining adequate items at strategic locations (i.e., reserves) are considerable inventory-related costs. Unpredicted demand leads to a high degree of uncertainty. Efforts to mitigate the uncertainty should rely not only on prepositioning supplies at reserves but also on integrating various channels of medical materials. This paper proposes multi-mitigation strategies in medical supplies to ensure uninterrupted supply for hospitals and significant savings by introducing two-type suppliers, reserving and manufacturing suppliers. Thus, each hospital with uncertain demand is enabled to be served by various channels during pandemics: prepositioning in reserves, backups served by reserving suppliers, and medical commodities produced by manufacturing suppliers. Stochasticity is also incorporated into the raw materials available to produce. This research aims to develop an emergency response application that integrates preparedness action (reserve location, inventory level, and contract supplier's selection) with post-event operations (allocating medical materials from various channels). We formulate a two-stage stochastic mixed integer program to determine prepositioning strategy, including two-type suppliers' selection, and post-event allocation of multiple sources. A branch-and-Benders-cut method is developed for this problem and significantly outperforms both the classical Benders decomposition and Gurobi in the solution time. Different-sized test instances also verify the robustness of the proposed method. Based on a realistic and typical case study (inspired by the COVID-19 pandemic in Wuhan, China), significant savings, an increase in inventory utilization and an increase in demand fulfilment are obtained by our approach.

Scenario-dominance to multi-stage stochastic lot-sizing and knapsack problems

This paper presents strong scenario dominance cuts for effectively solving the multi-stage stochastic mixed-integer programs (M-SMIPs), specifically focusing on the two most well-known M-SMIPs: stochastic capacitated multi-item lot-sizing (S-MCLSP) and the stochastic dynamic multi-dimensional knapsack (S-MKP) problems. Scenario dominance is characterized by a partial ordering of scenarios based on the pairwise comparisons of random variable realizations in a scenario tree of a stochastic program. In this paper, we study the implications of scenario-dominance relations and inferences obtained by solving scenario sub-problems to drive new strong cutting planes to solve S-MCLSP and S-MKP instances faster. Computational experiments demonstrate that our strong scenario dominance cuts can significantly reduce the solution time for such M-SMIP problems with an average of 0.06% deviation from the optimal solution. The results with up to 81 random variables for S-MKP show that strong dominance cuts improve the state-of-the-art solver solution of two hours by 0.13% in five minutes. The proposed framework can also be applied to other scenario-based optimization problems.

A Two-stage Stochastic Mixed-integer Programming Model for Resilience Enhancement of Active Distribution Networks

Most existing distribution networks are difficult to withstand the impact of meteorological disasters. With the development of active distribution networks (ADNs), more and more upgrading and updating resources are applied to enhance the resilience of ADNs. A two-stage stochastic mixed-integer programming (SMIP) model is proposed in this paper to minimize the upgrading and operation cost of ADNs by considering random scenarios referring to different operation scenarios of ADNs caused by disastrous weather events. In the first stage, the planning decision is formulated according to the measures of hardening existing distribution lines, upgrading automatic switches, and deploying energy storage resources. The second stage is to evaluate the operation cost of ADNs by considering the cost of load shedding due to disastrous weather and optimal deployment of energy storage systems (ESSs) under normal weather condition. A novel modeling method is proposed to address the uncertainty of the operation state of distribution lines according to the canonical representation of logical constraints. The progressive hedging algorithm (PHA) is adopted to solve the SMIP model. The IEEE 33-node test system is employed to verify the feasibility and effectiveness of the proposed method. The results show that the proposed model can enhance the resilience of the ADN while ensuring economy.

Green-Resilient Supplier Selection and Order Allocation Under Disruption by Utilizing Conditional Value at Risk: Mixed Response Strategies

Two important decisions in supply chains and logistics systems design are the supplier selection and order allocation (SS&OA) problem and the vehicle routing problem (VRP). Supply disruption may reduce the capacity of suppliers, and the transportation network disruption may decrease the number of vehicles in the fleet and disrupt some routes. Also, increasing environmental regulations and environmental awareness makes companies pay more attention to green supply chain management (GSCM). In this paper, we integrate green and resilient supplier selection and order allocation decisions with vehicle routing decisions under disruption. We present a multiproduct two-stage risk-averse mixed-integer stochastic linear programming for the green and resilient supplier selection and order allocation integrated with vehicle routing (G&RSS&OA-V) problem. We consider resilient strategies before disruption, including multiple sourcing, supplier fortification, prepositioned inventory at the protected supplier, and contract with third-party logistics providers (3PLs). The objective function is to minimize the total mean-risk cost and the cost of greenhouse emissions. We use conditional value at risk (CVaR) as a risk measure to control the risk of worst-case cost. The most significant decisions of this model are the strategic decisions of determining the optimal suppliers and the operational decisions of vehicles routing under disruption simultaneously. Other decisions include determining which suppliers should be fortified, the amount to be transported to the hybrid manufacturing-distribution (HMD) center through the supplier or prepositioned emergency inventory, and the amount of lost sales. In order to validate the proposed model and its features, several numerical examples along with sensitivity analysis are performed by GAMS software, which shows the efficiency and application of the developed model, and some managerial insights are reported. The results of the sensitivity analysis show that as α increases from 0.1 to 0.9, the mean-CVaR objective function cost increases to 13.2%. As λ increases from 0.1 to 0.9, the mean-CVaR objective function cost increases to 35.6%. The increase of these two risk factors makes the proposed model more risk-averse. As the expected shortage cost increases by 150%, the mean-CVaR objective function cost increases to 36% while the amount of expected shortage decreases by 56%.

Short-circuit constrained distribution network reconfiguration considering closed-loop operation

This paper presents a novel scenario-based stochastic mixed-integer second-order cone programming model to solve the problem of optimal reconfiguration of distribution systems with renewable energy sources considering short-circuit constraints. The proposed formulation minimizes technical losses by modifying the statuses of sectionalizing and tie switches, allowing the operation of distribution networks with radial and closed-loop topologies. Since the formation of loops could impact fault current levels, short-circuit constraints are considered in the problem formulation. Numerical experiments are carried out using an 84-node system and the results demonstrate the effectiveness of the proposed formulation to reduce technical losses notably when a closed-loop operation is allowed. Additionally, it is verified that short-circuit constraints prevent the adoption of network configurations with high short-circuit values.

A stochastic programming approach to surgery scheduling under parallel processing principle

Parallel processing is a principle which enables simultaneous implementation of anesthesia induction and operating room (OR) turnover with the aim of improving OR utilization. In this article, we study the problem of scheduling surgeries for multiple ORs and induction rooms (IR) that function based on the parallel processing principle under uncertainty. We propose a two-stage stochastic mixed-integer programming model considering the uncertainty in induction, surgery and turnover durations. We sequence patients and set appointment times for surgeries in the first stage and assign patients to IRs at the second stage of the model. We show that an optimal myopic policy can be used for IR assignment decisions due to the special structure of the model. We minimize the expected total cost of patient waiting time, OR idle time and IR idle time in the objective function. We enhance the model formulation using bounds on variables and symmetry-breaking constraints. We implement a novel progressive hedging algorithm by proposing a penalty update method and a variable fixing mechanism. Based on real data of a large academic hospital, we compare our solution approach with several scheduling heuristics from the literature. We assess the additional benefits and costs associated with the implementation of parallel processing using near-optimal schedules. We examine how the benefits are inflated by increasing the number of IRs. Finally, we estimate the value of stochastic solution to underline the importance of considering uncertainty in durations.