Two-stage Mixed-integer Programming Model Research Articles

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.

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.

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.

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.

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.

Developing a resilient supply chain through supplier flexibility and reliability assessment

In this study, we examine the optimal allocation of demand across a set of suppliers in a supply chain that is exposed to supply risk and environmental risk. A two-stage mixed-integer programming model is used to develop a flexible sourcing strategy under disruptions. Our model integrates supplier selection and demand allocation with transportation channel selection and provides contingency plans to mitigate the negative impacts of disruptions and minimise total network costs. Finally, a numerical example is presented to illustrate the model and provide insights. The findings suggest that developing contingency plans using flexibility in suppliers’ production capacity is an effective strategy for firms to mitigate the severity of disruptions. We also show that flexibility and reliability of the suppliers and regions play a significant role in determining contingency plans for during disruption. Findings generally show that highly flexible suppliers receive less allocation, and their flexible capacity is reserved for disruptions. For firms that do not incorporate risk management into supplier selection and allocation, the recommendation is to source from fewer, more reliable suppliers with less risk of disruption. Our findings also emphasise that the type of disruption has important implications for supplier selection and demand allocation. This study highlights the supply chain risk management strategy of regionalising as a means for minimising the impact of environmental disruptions.

An affine adjustable robust model for generation and transmission network planning

This work studies an electricity generation and transmission network planning problem where loads and cost parameters are uncertain. The problem is to first determine the generation and transmission capacities to install in the supply network. When the uncertainties are revealed, a flow plan is developed to minimize the total costs and to balance loads. A two-stage mixed-integer programming model is proposed to maximize the robustness of the plan in achieving a total cost budget target. The modeling approach in this study synthesizes recent developments in affine adjustable robust optimization technology and decision-making behavior under uncertainty. A novel solution approach is also proposed to achieve a safe and tractable approximation of the model. This involves the partitioning of the total cost budget target in order to transform the original problem into a small collection of mixed integer programming models that can be solved efficiently using standard mixed integer programming solvers. Numerical studies using a power generation network are performed, which demonstrate that the proposed robust planning model performs favorably compared to a stochastic programming model across different performance measures. The computational results strongly suggest the ability of the robust planning model to effectively mitigate the effect of uncertainties.