Two-stage Mixed-integer Programming Research Articles

A Two-Stage Robust Pricing Strategy for Electric Vehicle Aggregators Considering Dual Uncertainty in Electricity Demand and Real-Time Electricity Prices

To enable the regulation and utilization of electric vehicle (EV) load resources by the power grid in the electricity market environment, a third-party electric vehicle aggregator (EVA) must be introduced. The strategy of EVA participation in the electricity market must be studied. During operation, the EVA faces a double uncertainty in the market, namely, electricity demand and electricity price, and must optimize its market behavior to protect its own interests. To achieve this goal, we propose a robust pricing strategy for the EVA that takes into account the coordination of two-stage market behavior to enhance operational efficiency and risk resistance. A two-stage robust pricing strategy that takes into account uncertainty was established by first considering day-ahead pricing, day-ahead electricity purchases, real-time electricity management, and EV customer demand response for the EVA, and further considering the uncertainty in electricity demand and electricity prices. The two-stage robust pricing model was transformed into a two-stage mixed integer programming by linearization method and solved iteratively by the columns and constraints generation (CCG) algorithm. Simulation verification was carried out, and the results show that the proposed strategy fully considers the influence of price uncertainty factors, effectively avoids market risks, and improves the adaptability and economy of the EVA’s business strategy.

Collaborative optimization of depot location, capacity and rolling stock scheduling considering maintenance requirements

Generally, when optimizing a rolling stock schedule, the locations of the depots, or places in the network where the composition changes and maintenance occurs, are assumed known. The locations where maintenance is performed naturally influence the quality of any resulting rolling stock schedules. In this paper, the problem of selecting new depot locations and their corresponding capacities is considered. A two-stage mixed integer programming approach for rolling stock scheduling with maintenance requirements is extended to account for depot selection. First, a conventional flow-based model is solved, ignoring maintenance requirements, to obtain a variety of rolling stock schedules with multiple depot locations and capacity options. Then, a maintenance feasible rolling stock schedule can be obtained by solving a series of assignment problems by using the schedules found in the first stage. The proposed methodology is tested on real-life instances, and the numerical experiments of different operational scenarios are discussed.

Developing a national pandemic vaccination calendar under supply uncertainty

During the COVID-19 pandemic, many countries faced challenges in developing and maintaining a reliable national pandemic vaccination calendar due to vaccine supply uncertainty. Deviating from the initial calendar due to vaccine delivery delays eroded public trust in health authorities and the government, hindering vaccination efforts. Motivated by these challenges, we address the problem of developing a long-term national pandemic vaccination calendar under supply uncertainty. We propose a novel two-stage mixed integer programming model that considers critical factors such as multiple doses, varying dosing schemes, and uncertainties in vaccine delivery timing and quantity. We present an efficient aggregation-based algorithm to solve this complex problem. Additionally, we extend our model to allow for dynamic adjustments to the vaccine schedule in response to mandatory policy changes, such as modifications to dose intervals, during ongoing vaccinations. We validate our model based on a case study developed by using real COVID-19 vaccination data from Norway. Our results demonstrate the advantages of the proposed stochastic optimization approach and heuristic algorithm. We also present valuable managerial insights through extensive numerical analysis, which can aid public health authorities in preparing for future pandemics.

Location–allocation problem in the emergency logistics system considering lateral transshipment strategy

This paper explores the problem of designing an efficient emergency logistics system while considering lateral transshipment strategy. Firstly, a two-stage mixed-integer programming model is proposed to characterize the location–allocation problem in an emergency logistics system with the aim of minimizing the total cost. Secondly, we propose an improved Benders decomposition framework for solving the emergency logistics system model. The numerical results reveal that our algorithm can find the optimal solution efficiently for large-scale instances, outperforming the commercial programming solver Gurobi. Finally, using Poyang Lake in China as a case study, we demonstrate that the lateral transshipment strategy can reduce the total cost of the emergency logistics system by 8.3% through comparing with the one without considering the lateral transshipment strategy. At the same time, we argue that the equity factor prevents the affected people from being evacuated to more distant shelters. The contribution of our work can be summarized as follows: an improved Benders decomposition algorithm is developed to investigate the optimal design of location–allocation strategy in the three-level emergency logistics network, with the consideration of lateral transshipment strategy and equity factor.

Computation of Systemic Risk Measures: A Mixed-Integer Programming Approach

Mixed-Integer Programming Formulations for Systemic Risk Measures Measurement and allocation of systemic risk have become important tasks in interconnected financial networks. In “Computation of Systemic Risk Measures: A Mixed-Integer Programming Approach,” Ararat and Meimanjan consider a clearing model that considers external assets and business costs of a bank simultaneously by extending the Rogers–Veraart network model with default costs. They prove that clearing payment vectors can be calculated by solving mixed-integer programming problems in which the existence of business costs and default costs are represented by binary variables. The systemic risk measure corresponding to this model yields nonconvex sets of capital allocation vectors. The authors compute these nonconvex sets by solving a multiobjective optimization problem whose scalarizations are two-stage mixed-integer programming problems with an expectation constraint. The computational procedure is versatile enough to accommodate other applications of systemic risk where the risk factors are aggregated by solving a general mixed-integer programming problem.

Blocks allocation and handling equipment scheduling in automatic container terminals

The blocks allocation and integrated scheduling of quay cranes (QCs), yard cranes(YCs), and automated guided vehicles(AGVs) between the apron, side-loading blocks and end-loading blocks for multiple berthing ships is investigated, where the inbound, outbound and transship containers from several ships berthing at the same time is considered. A multi-ship blocks-sharing strategy is proposed to increase the efficiency in double cycling operations of YCs and AGVs through centrally storing containers from different ships. A two-stage mixed integer programming model is formulated in which the first stage is to optimize the problem of blocks allocation and QCs scheduling, and the second stage is to optimize the problem of YCs and AGVs scheduling. To solve the model, adaptive variable neighbourhood immune algorithm (AVNIA) and rule-based heuristic algorithm are proposed. Compared with the lower bound compiled with Gurobi, ANVIN is highly effective based on case study layout data of the automated container terminal of Yangshan port, Shanghai. Experiments demonstrate that the proposed blocks-sharing strategy can significantly improve the utilization rate of YCs and AGVs, and effectively reduce the operation cost of automatic container terminals.

Adaptive distributionally robust hub location and routing problem with a third-party logistics strategy

In this paper, we study an essential and more realistic variant of the hub location-routing problem in which third-party logistics (3PL) are incorporated into a self-built distribution hub network. A two-stage mixed-integer programming model is developed to address the proposed hub location-routing problem with mixed logistical strategies. In the strategic stage, the hub location and allocation decisions are made before the realization of uncertain demands to minimize the total construction and transportation costs. In the operational stage, the managers make vehicle routing decisions with and without the 3PL strategy, and route planning is considered for multiple owned and 3PL vehicles on the basis of simultaneous pickup and delivery. We investigate the effects of uncertain factors on the network design of this framework, as the node demands are often unobservable or difficult to obtain accurately in practice. To address the inherent uncertainty of the problem, we propose an adaptive distributionally robust model by limiting uncertain demands to a specified ambiguity set. Moreover, we reformulate the proposed model into tractable robust counterpart forms under the ambiguity set with information about the support, mean and upper bounds on the dispersion. We design a genetic algorithm with local search to solve the reformulated model on a large scale. Finally, we conduct a case study for a leading retail enterprise in the Beijing–Tianjin–Hebei region to demonstrate the effectiveness and applicability of our model.

Joint Optimization of Pre-Marshalling and Yard Cranes Deployment in the Export Block

To improve the efficiency of loading operation by researching the optimization of the pre-marshalling operation scheme in the export container block between the time when the ship stowage chart was published and the beginning time of loading, a two-stage mixed integer programming model was established. The first stage established an optimization model of the container reshuffling location, based on the objective function of the least time-consuming operation of a single-bay-yard crane, and designed an improved artificial bee colony algorithm to solve it. Based on the first stage, an optimization model of yard crane configuration and scheduling was built to minimize the maximum completion time of the yard crane in the export block, and an improved genetic algorithm was designed to solve the built model. Through comparative analysis, the performance of our algorithm was better than CPLEX and traditional heuristic algorithms. It could still solve the 30 bays quickly, and the solving quality was 8.53% and 11.95% higher than GA and TS on average, which verified the effectiveness of the model and the science of the algorithm and could provide a reference for improving the efficiency of port operation.

Integrated proactive-reactive approach and a hybrid adaptive large neighborhood search algorithm for berth and quay crane scheduling under uncertain combination

<p style='text-indent:20px;'>The integrated berth and quay crane scheduling problem is frequently encountered in port management issue, and the dynamic random data obtained by the container terminals under uncertain combination usually makes it unworkable to execute the scheduling. To promote sustainability and adaptivity of container terminals, this paper focuses on developing an integrated proactive-reactive approach for berth and quay crane scheduling under the fluctuation of vessels' arrival time and breakdown of quay crane. The problem is modelled as a two-stage mixed integer program where the baseline schedule with robustness is established in the first stage, and the recovery schedule is established in the second stage. This study aims to minimize the total management cost deviation and total time deviation of the terminal scheduling, so as to implement minimal changes compared with the baseline schedule. Therefore, the scheduling performance such as robustness, recoverability and adaptability is considered. To solve this model, a hybrid adaptive large neighborhood search (HALNS) algorithm framework combined with variable neighborhood search is developed, and the greedy algorithm is developed to construct the initial feasible solution. The performance evaluation tests of the proposed algorithm are performed on the three scales of benchmark. Computational results indicate the strength of solution methods and effectiveness of the proposed model.</p>

Robust optimization of train scheduling with consideration of response actions to primary and secondary risks.

Nowadays, with the rapid development of rail transportation systems, passenger demand and the possibility of the risks occurring in this industry have increased. These conditions cause uncertainty in passenger demand and the development of adverse impacts as a result of risks, which put the assurance of precise planning in jeopardy. To deal with uncertainty and lessen negative impacts, robust optimization of the train scheduling problem in the presence of risks is crucial. A two-stage mixed integer programming model is suggested in this study. In the first stage, the objective of the nominal train scheduling problem is to minimize the total travel time function and optimally determine the decision variables of the train timetables and the number of train stops. A robust optimization model is developed in the second stage with the aim of minimizing unsatisfied demand and reducing passenger dissatisfaction. Additionally, programming is carried out and the set of optimal risk response actions is identified in the proposed approach for the presence of primary and secondary risks in the train scheduling problem. A real-world example is provided to demonstrate the model's effectiveness and to compare the developed models. The results demonstrate that secondary risk plays a significant role in the process of optimal response actions selection. Furthermore, in the face of uncertainty, robust solutions can significantly and effectively minimize unsatisfied demand by a slightly rise in the travel time and the number of stops obtained from the nominal problem.

Intermodal transportation hub location optimization with governments subsidies under the Belt and Road Initiative

Driven by globalization, the COVID-19 outbreak has severely impacted global transport and logistics systems. To better cope with this globalization crisis, the Belt and Road Initiative (BRI)—based on the concept of cooperation—is more important than ever in the post-pandemic era. Taking the BRI as the background, we design an intermodal hub-and-spoke network to provide reference for governments along BRI routes to improve their cross-border transportation system and promote economic recovery. In the context of the BRI, local governments at different nodes have incentives to subsidize hub construction and/or rail transportation to boost economic development. We consider co-opetition behavior among different levels of government caused by subsidies in this intermodal hub location problem, which we call the intermodal hub location problem based on government subsidies. We establish a two-stage mixed-integer programming model. In the first stage, local governments provide subsidies, then the central government decides the number and location of hubs. In the second stage, freight carriers choose the optimal route to transport the goods. To solve the model, we design an optimization method combining a population-based algorithm using contest theory. The results show that rail subsidies are positively correlated with construction subsidies but are not necessarily related to the choice of hubs. Compared with monomodal transportation, intermodal transportation can reduce costs more effectively when there are not too many hubs and the cost of different modes of transportation varies greatly. The influences of local government competition and hub construction investment on network design and government subsidies are further examined.

Multi-Objective Parallel Machine Scheduling with Eligibility Constraints for the Kitting of Metal Structural Parts

This paper studied a class of coupling problems of material assignment, part nesting, kit delivery and parallel machine scheduling. The aim of this paper was to solve the scheduling problem of metal structural part processing and welding assembly with eligibility constraints. A two-stage mixed-integer programming model was constructed. The eligibility constraints took into account the material type of parts and nesting. The objectives were to minimize the makespan, maximize material utilization and minimize the kit delivery metrics (kitting time and numbers of earliness and tardiness of kits). A hierarchical optimization approach was proposed. The scheduling model was solved by using the Gurobi solver in the first stage, and the results were used to constrain the second stage. The second stage of the scheduling model was solved using an improved multi-objective genetic algorithm. Due to the strong coupling relationships among the sorting of parts, the sorting of each profile and the sorting of each material, a hybrid encoding and decoding mode was designed for part sorting with eligibility constraints. Finally, the proposed scheduling approach was applied to actual production cases. The data showed that when the number of components exceeded 300 (the number of parts was about 1500), the material utilization reached 95%. Choosing a suitable number of machines, machine utilization reached 90%. The results demonstrated the effectiveness of the proposed scheduling model and algorithm.

Two-Stage Robust Optimization Model for Uncertainty Investment Portfolio Problems

Investment portfolio can provide investors with a more robust financial management plan, but the uncertainty of its parameters is a key factor affecting performance. This paper conducts research on investment portfolios and constructs a two-stage mixed integer programming (TS-MIP) model, which comprehensively considers the five dimensions of profit, diversity, skewness, information entropy, and conditional value at risk. But the deterministic TS-MIP model cannot cope with the uncertainty. Therefore, this paper constructs a two-stage robust optimization (TS-RO) model by introducing robust optimization theory. In case experiments, data crawler technology is used to obtain actual data from real websites, and a variety of methods are used to verify the effectiveness of the proposed model in dealing with uncertainty. The comparison of models found that, compared with the traditional equal weight model, the investment benefits of the TS-MIP model and the TS-RO model proposed have been improved. Among them, the Sharpe ratio, Sortino ratio, and Treynor ratio have the largest increase of 19.30%, 8.25%, and 7.34%, respectively.

Benders decomposition with integer sub-problem applied to pipeline scheduling problem under flow rate uncertainty

Important issues in a pipeline system are energy efficiency, reliability and throughput flexibility. Practically conventional pumps are not capable of operating at the highest attainable efficiency for long running time. This deficiency has prevented a pipeline system to operate close to its predefined program. A possible remedy is to take into account uncertainty due to pumps operations. In this paper, a stochastic two-stage mixed integer programming (MILP) model is developed for the multiproduct pipeline-scheduling problem under flow rate uncertainty. The problem arises in a number of settings, and the real-world applicability discussed and demonstrated. The stochastic MILP model involves many discrete variables that make it intractable for real-life cases. As a solution method, the sample average approximation is combined with a three-step solution approach based on Benders decomposition. The modeling and solving approach is evaluated in some case studies including a real-life problem from NIOPTC.

Robust Optimization of Airplane Passenger Seating Assignments

We present a method that reduces the time it takes to complete the passenger boarding of an airplane. In particular, we describe a two-stage mixed integer programming (MIP) approach, which assigns passengers to seats on an airplane based on the number of bags they carry aboard the plane. The first stage is an MIP that assigns passengers to seats to minimize the time to complete the boarding of the plane. The second-stage MIP also determines seating assignments, while constraining the total boarding time to that determined by the stage-one MIP and maximizing weighted slack times to provide a more robust assignment. Numerical results show that this two-stage approach results in lower average boarding times than the one-stage approach, when the time it takes passengers to walk and sit in their seats is random. Experiments indicate that the magnitude of the improvement is not very sensitive to variations in the slack time weights.

Enhanced index tracking in portfolio optimization with two-stage mixed integer programming model

Enhanced index tracking is a portfolio management which aims to construct the optimal portfolio to generate higher return than the benchmark index return at minimum tracking error without purchasing all the stocks that make up the index. The objective of this paper is to propose a two-stage mixed integer programming model to improve the existing single-stage mixed integer programming model for tracking FBMKLCI Index in Malaysia. The optimal portfolio performance of both models are determined and compared in terms of portfolio mean return, tracking error, excess return and information ratio. The results of this study indicate that the optimal portfolio of the proposed model generates weekly excess return over the benchmark FBMKLCI index return at minimum tracking error. Besides that, the proposed model is able to outperform the existing model in tracking the benchmark index.

Optimization of Water Resource Management Using Chooser Option Contracts under Uncertainty

Optimization of water resource management is very important to reduce water shortage risk and increase economic benefit. Water option can be used as a flexible instrument in water resource management. In this study, chooser options are designed for water managers to buy additional water in drought periods and sell redundant water in wet periods. To make optimal decisions under uncertainties including probability distributions and intervals, an inexact two-stage mixed-integer programming (ITSMIP) model is applied. To illustrate the role of chooser options in water resource management, a case study is presented. Factors which influence the willingness of buying the chooser options are analyzed and the optimal water allocation plans are also provided. The results demonstrate that chooser options can help water managers mitigate the impact of water shortage and increase utilization efficiency of water resource greatly.

Optimization of water allocation decisions under uncertainty: the case of option contracts

For decades, water managers need to allocate water efficiently to competing users due to growing water demand and shortage. Water option contract is a flexible management tool for water managers. In this study, with consideration of the uncertainties of probability distributions and intervals, an inexact two-stage mixed-integer programming (ITSMIP) model is applied to support the decision making of water allocation when option contracts are proposed. A case study of water allocation with the option contract is presented. Factors which influence the willingness of signing the option contract are analysed and the optimum water allocation plans are also provided. The results demonstrate that water option contracts can help water managers mitigate the impact of water shortage and increase economic benefits largely.

An assessment of supply chain disruption mitigation strategies

We assess the effectiveness of incorporating three types of redundancy practices (pre-positioning inventory, backup suppliers, and protected suppliers) into a firm’s supply chain that is exposed to ttwo types of risk: supply risk and environmental risk. Supply risk disrupts an individual supplier, while environmental risk makes a number of suppliers in a given region unavailable. An additional factor is supplier interdependence, where a disruption in one supplier may also disrupt other active suppliers. Utilizing the concept of a decision tree to capture different disruption scenarios, we develop a two-stage mixed-integer programming (two-stage MIP) model as a General Model to address the problem of supplier selection and order allocation under supplier dependencies and risk of disruptions. In the General Model, multi-sourcing is the only supplier strategy that the firm implements. Then we develop three separate extensions of the General Model, one for each of the three redundancy practices, and evaluate the expected supply chain cost of each extended model. We quantitatively show how adding redundancy to the supply chain in different forms, along with contingency plans, can help firms mitigate the impact of supply chain disruptions. The findings suggest that all three strategies reduce costs and risks compared to the General Model. An analysis of reliability, risks, dependence, and costs is conducted on each strategy to provide insights into supplier selection, demand allocation, and capability development in a supply chain under supply chain risks. Finally, we show that regionalizing a supply chain is an effective way to mitigate the negative impacts of environmental disruptions on the supply chain.

Sensor-Driven Condition-Based Generator Maintenance Scheduling—Part II: Incorporating Operations

A framework for sensor driven condition based generator maintenance scheduling was proposed in Part I of this paper. In Part II, we extend the previous model by incorporating the unit commitment and dispatch into the optimal maintenance scheduling problem. We reformulate this extended maintenance scheduling problem as a two-stage mixed integer program. We use this reformulation to construct an algorithm that obtains the global optimal solution to the proposed generator maintenance problem. Finally, we test and analyze the proposed model through extensive experiments conducted on IEEE-118 bus system. For every experiment, we present a benchmark analysis against the maintenance models used in current industry practice and power systems literature. Experimental results indicate that the proposed maintenance schedules provide considerable improvements in both cost and reliability.