Two-stage Mixed-integer Linear Programming Research Articles

Assessing the Impact of Preprocessing and Conversion Technologies on the Sustainable Aviation Fuel Supply from Forest Residues in the Southeast U.S.A.

Forest residues have been suggested as potential feedstock for sustainable aviation fuel (SAF) production through different pathways. However, the bulky density and scatter distribution require efficient size-reduction processes to enhance the feedstock supply. Thus, this study analyzes the impact of alternative preprocessing and conversion technologies on the SAF supply chains from logging residues (the predominant share of forest residues) using a two-stage mixed-integer linear programming model. The model determines the optimal location of biomass preprocessing depots, conversion facilities, and airports receiving SAF by minimizing the net present value of the total supply chain cost over 10 years. Using high-resolution spatial data in the Southeast U.S.A. as a case study, the results show that the technology and scale of the facilities heavily influenced the SAF’s breakeven cost and maximum supply quantity. The most economically efficient system that adopts a rotary shear milling system and pyrolysis conversion process could generate up to 650 million gallons of SAF from logging residues in the region, with nearly 62% of the SAF being produced from hardwood residues. Also, the most efficient system has an estimated breakeven cost of US$5.51 per gallon of SAF.

Optimizing high-speed railway express system under uncertainty

The rapid expansion of high-speed railway (HSR) networks has elevated the HSR express system, which combines road and HSR transport, as a notable intermodal transport choice. This research delves into the optimization challenges present in the HSR express system's operation, given the unpredictable freight demands. These challenges encompass vehicle arrangement, station selection, freight allocation, and the optimization of HSR freight routes. To address these challenges, we introduce a two-stage mixed integer linear programming model aiming to optimize the anticipated net gains of the HSR express system. We also put forward a meta-heuristic method to efficiently solve the model. Empirical tests and sensitivity studies, grounded in a real-world example from the China Railway Nanchang Group Co., Ltd., are executed. Additionally, advanced techniques, including parallel computing tailored to the problem's nature, have been employed to trim down computational times, especially for intricate real-world scenarios. The findings from this study provide valuable insights for industry practitioners.

Joint operation of mobile battery, power system, and transportation system for improving the renewable energy penetration rate

Future energy system will feature in a high-share of renewable energies (REs), which poses huge challenges to obtain full utilization of renewable power generation. To solve the problem, this paper presents a joint-operation two-stage mixed integer linear programming model to coordinate the power system and train transportation system by optimizing the logistics of mobile batteries and real-time charge/discharge in cities. The full/empty batteries are transported through the train transportation system between the load side and the renewable energy station, which improves renewable energy penetration, economics, and mobilities. The first-stage model deals with the preliminary logistic optimization of batteries including routes, time and capacities of transportation, and the second-stage model optimizes real-time schedule of charge/discharge of batteries. The aim of the proposed model is to minimize the transportation cost, maximize the utilization rate of renewable energy in an energy system and smooth the daily load curve. The proposed model considers technical constraints such as railway transportation capacity, load demand satisfaction and renewable energy consumption in the power system. The optimal logistics plan and real-time charging and discharging plan can be obtained for both full and empty battery transportation. The validity of the method is verified with real data from Northeast China and Northern China, including railway routes, renewable energy output, and load profiles. The results show that the renewable energy penetration rate increases from 84.4% to 95.3% in Northeast China and from 42.7% to 78.2% in Northern China under the 2050 scenario with the proposed method. The peak-to-valley difference in the daily load curve decreases by 84% and 100% in Northeast China and Northern China, respectively. Meanwhile, the case study shows that the costs of battery transportation decrease from 0.398 CNY/kWh and 0.377 CNY/kWh to 0.252 CNY/kWh and 0.254 CNY/kWh when energy density of battery increases from 0.170 kWh/kg to 0.250 kWh/kg. This indicates that mobile energy storage has great economics. Moreover, renewable energy penetration rate is increased and peak-to-valley difference of daily load curve is decreased with no additional transmission lines.

A bi-objective optimization approach for carbon capture and storage supply chain network combining with pricing policies: Economic and social aspects

Carbon capture and storage (CCS) technology and pricing policies are two mitigation strategies that attempt to tackle the carbon emission problem. Nonetheless, the uncertainty of public reaction to CCS infrastructure installation has been noted as a significant issue that may lead to either cancelations, delays, or influence the deployment cost of CCS infrastructure. Although these strategies are solutions to reduce carbon emissions, the entire supply chain itself needs to align with sustainability policy. While some previous studies have addressed mitigation strategies or the combination of social objective and CCS technology, the impact of social objectives on reservoir openness has not been explored. This article contributes to the literature by illustrating the effect of social objectives on reservoir openness and the total cost of the CCS supply chain. Through the development of a two-stage mixed-integer linear programming model, we optimize the supply chain network design. In the first stage, a single-objective model minimizes capture, transportation, and sequestration costs along with CO2 emission costs to the atmosphere, considering various carbon tax prices. In the second stage, a bi-objective model maximizes social acceptance towards CO2 injection wells (reservoirs) to analyze the impact of cultural dimensions. Our findings reveal that maximizing the social objective leads to increased costs compared to the economic optimum, and it significantly influences the openness of reservoirs based on cultural dimensions. Furthermore, we demonstrate that an increase in the CO2 tax enhances the amount of captured carbon and the complexity of the pipeline network in the supply chain. This insight has implications for governments and decision-makers who can encourage industries to adopt CCS technology by setting appropriate carbon tax prices. To validate the model, we apply it to a case study in the European Union cement industry, one of the major carbon emitters among all industries. Our results not only confirm the model's effectiveness but also show its potential for practical application. Overall, our study presents a comprehensive framework for designing a two-stage sustainable supply chain network for CCS technology that optimizes economic and social objectives while considering CO2 pricing.

Two-stage hybrid model for supplier selection and order allocation considering cyber risk

In the context of collaborative manufacturing, cyber risk caused by cyber attacks may lead to severe supply chain disruption. Currently, supplier selection and order allocation is regarded as effective means to mitigate the risks that might cause disruption. Thus, we propose a two-stage hybrid model for supplier selection and order allocation under cyber risk. The hybrid model consists of fuzzy analytical hierarchy process (Fuzzy AHP), Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), and two-stage mixed integer linear programming (MIP). Based on the extracted cyber risk indicators, a Fuzzy AHP is used to calculate the level of cyber risk of suppliers. TOPSIS is utilized to quantitatively evaluate the cyber risk of suppliers and determine the ranking of suppliers. Then, a two-stage MIP model is developed to support decision-making on order allocation. The first-stage decisions are determined without emergencies, and the second-stage decisions are determined under emergencies. The results reveal that application of the proposed two-stage hybrid model could mitigate the negative impacts of cyber risks. By providing a theoretical basis and quantitative method for cyber risk evaluation, this research is of theoretical and practical significance to the field of supply chain management.

Elective surgery scheduling considering transfer risk in hierarchical diagnosis and treatment system

With the aggravation of the shortage of staffing hospital beds, elective surgeries have to be postponed or even cancelled, which directly affects hospital income and patient health. Therefore, we propose a fuzzy scheduling model based on a patient transfer strategy in the hierarchical diagnosis and treatment system to ensure timely surgery. We propose a risk estimation method based on health failure mode and effect analysis to reduce the transfer risk and represent the surgery duration and the length of stay as fuzzy numbers to deal with the uncertainty. Given that solving the fuzzy model is challenging, we propose an equivalent formulation to transform the fuzzy model into a two-stage mixed integer linear programming (TMIP) model, which can reduce the loss of decision-making information. Finally, the column and constraint generation algorithm is used to solve the TMIP model that is suited for the structure of the main problem and subproblem. The experiment shows that the patient transfer strategy can effectively relieve the hospital bed shortage, and its potential can be further released if the transfer risk can be properly assessed. The proposed method for solving the fuzzy model can reduce the information loss comparing traditional methods.

Storage Allocation in Active Distribution Networks Considering Life Cycle and Uncertainty

The modern active distribution systems necessitate integrating storage systems, thereby facilitating the large-scale proliferation of photovoltaic (PV) energy resources. This further calls for the optimal planning of energy storage systems, satisfying all the operational and economic constraints. This article describes an exhaustive storage integration method, deeming the life cycle of the battery energy storage, the uncertainty of load and PV output, and the islanded mode of operation of the system. A two-stage mixed-integer linear programming problem is formulated that determines the capacity and the number of discharge cycles of the batteries in the first stage. The lifetime of the battery based on the partial depth of discharge is analyzed in the second stage. Furthermore, the uncertainty and variability of PV and demand are taken into account through probabilistic analysis and time-period clustering. The method is validated on a standard 33-bus radial distribution network for the allocation of distributed lithium-ion batteries. Also, the method’s scalability is validated on a practical Indian distribution network and a 141-bus distribution network of metropolitan area of Caracas with distributed PV installations on various nodes.

Risk-averse based optimal operational strategy of grid-connected photovoltaic/wind/battery/diesel hybrid energy system in the electricity/hydrogen markets

The techno-economic advantages of grid-connected hybrid energy system (HES) exploit synergies to improve reliability and economic efficiency while maintaining grid stability. Therefore, this paper proposes a risk-averse optimal operational strategy of grid-connected photovoltaic/wind/battery/diesel HES to participate into two energy markets including electricity and hydrogen markets. The grid company can flexibly trade power into two markets to maximally achieve profits based on price arbitrage. The risk influences of the uncertainties, i.e., photovoltaic/wind generation, and electricity prices on the expected revenue are evaluated with CVaR model. For a better exhibition of seasonal variability effects on HES optimal operation strategy, two typical Spring/Summer days are chosen. The proposed risk-averse optimal operational strategy is formulated as a two-stage mixed-integer linear programming (MILP) model. The results in a Spring day simulation under non-risk situation indicate that the overall expected revenue can be improved 2.74 times larger if considering hydrogen market. Moreover, the optimal operational strategy of hydrogen production is considerably affected by unpredictable wind farm. Sensitivity analysis also validates that the changes of PV/WT curtailment penalty have a profound influence than battery degradation coefficient on the HES expected revenue.

Optimal Covid-19 vaccine stations location and allocation strategies

PurposeThis study proposes strategies for vaccine center allocation for coronavirus disease (COVID) vaccine by determining the number of vaccination stations required for the vaccination drive, location of vaccination station, assignment of demand group to vaccination station, allocation of the scarce medical professional teams to station and number of optimal days a vaccination station to be functional in a week.Design/methodology/approachThe authors propose a mixed-integer nonlinear programming model. However, to handle nonlinearity, the authors devise a heuristic and then propose a two-stage mixed-integer linear programming (MILP) formulation to optimize the allocation of vaccination centers or stations to demand groups in the first stage and the allocation of vaccination centers to cold storage links in the second stage. The first stage optimizes the cost and average distance traveled by people to reach the vaccination center, whereas the second stage optimizes the vaccine’s holding and storage and transportation cost by efficiently allocating cold storage links to the centers.Findings The model is studied for the real-world case of Chandigarh, India. The results obtained validate that the proposed approach can immensely help government agencies and policymaking body for a successful vaccination drive. The model tries to find a tradeoff between loss due to underutilized medical teams and the distance traveled by a demand group to get the vaccination.Originality/value To the best of our knowledge, there are hardly any studies on a vaccination program at such a scale due to sudden outbreaks such as Covid-19.

Two-Stage MILP Model for Optimal Skeleton-Network Reconfiguration of Power System for Grid-Resilience Enhancement

Skeleton-network reconfiguration is an important task and plays a most important role during power system restoration (PSR) after a blackout. By determining the skeleton network, the power system can be restored as soon as possible to minimize the burden of network reconfiguration. In this paper, a resilience-based skeleton-network reconfiguration (NR) strategy, i.e., a two-stage mixed-integer linear programming (MILP)–based NR strategy for grid-resilience enhancement, is proposed to minimize the impact of emergency power outages on power systems. The start-up sequence for non-black-start generators (NBSGs) and line energization sequences are determined in the first-stage optimization model. A serial restoration constraint and a transient frequency constraint are considered to achieve the serial restoration scheme of NBSGs. Except for the generator power, all the variables attained during the first stage are assumed fixed in the second stage. The second-stage optimization model determines the optimal skeleton network by determining the target transmission lines and critical load pickup during the NR phase. Furthermore, the sets of metrics (i.e., system flexibility, power loss ratio, and recovery time resilience) are presented to determine the grid operational resilience in the skeleton-network reconfiguration. Then, two stages of skeleton-NR are formulated as MILP, and the branch-and-bound method is presented to solve both models. Finally, simulation studies are performed on the two modified power system test cases and the results validate that the proposed strategy can efficiently determine the robust skeleton network for practical and detailed PSR planning.

Distributed adaptive expansion approach for transmission and distribution networks incorporating source-contingency-load uncertainties

The plurality of active distribution networks (DNs) has made it ready to connect the transmission network (TN) for the dynamic power interaction. Coordinated expansion of transmission and distribution networks is the key to the economic and reliable operation of the integrated system in the future uncertain environment. This paper proposes a distributed decision-making framework to determine the robust TN expansion solution and stochastic DN expansion solutions. Uncertainty sets are deployed to model the uncertainties in TN while the scenario-based technique is implemented to tackle the uncertainties in DNs. To guarantee the consistency of transmission-distribution interaction, the deviation penalties of the interactive power are introduced into the objective function of each level. The iteration solution algorithm is used to transform the robust min–max-min problem for transmission expansion into a two-stage mixed-integer linear programming problem, and coordinate transmission and distribution expansion problems in a parallel manner. Case studies validate the effectiveness and high performance of the proposed distributed expansion method and the solution procedure.

A Mixed Epistemic-Aleatory Stochastic Framework for the Optimal Operation of Hybrid Fuel Stations

The fast development of technologies in the smart grids provides new opportunities such as co-optimization of multi-energy systems. One of the new concepts that can utilize multiple energy sources is a hybrid fuel station (HFS). For instance, an HFS can benefit from energy hubs, renewable energies, and natural gas sources to supply electric vehicles along with natural gas vehicles. However, the optimal operation of an HFS deals with uncertainties from different sources that do not have similar natures. Some may lack in term of historical data, and some may have very random and unpredictable behavior. In this study, we present a stochastic mathematical framework to address both types of these uncertainties according to the innate nature of each uncertain variable, namely: epistemic uncertainty variables (EUVs) and aleatory uncertainty variables (AUVs). Also, the imprecise probability approach is introduced for EUVs utilizing the copula theory in the process, and a scenario-based approach combining Monte Carlo simulation with Latin Hypercube sampling is applied for AUVs. The proposed framework is employed to address the daily operation of a novel HFS, leading to a two-stage mixed-integer linear programming problem. The proposed approach and its applicability are verified using various numerical simulations.

Genetic algorithm and Monte Carlo simulation for a stochastic capacitated disassembly lot-sizing problem under random lead times

The purpose of this research is to propose several optimization methods for the stochastic multi-period disassembly lot-sizing problem. The case of one type of end-of-life product and a two-level disassembly system is studied. The disassembly lead times are discrete random variables with a known and bounded probability distribution. The objective is to optimize the expected value of the total cost, which is the sum of setup cost, overload cost, inventory holding cost and backlogging cost. Three approaches were developed to solve the studied problem: (i) a two-stage mixed-integer linear programming model based on all possible scenarios for small instances, (ii) a sample average approximation approach based on Monte Carlo simulation for medium-scale instances and, (iii) an optimization approach based on the Monte Carlo simulation and a genetic algorithm for large-scale instances. Experimental results show the effectiveness of the proposed models which can be used to support decision-making on replenishment and disassembly plans.

A robust optimization model for sustainable and resilient closed-loop supply chain network design considering conditional value at risk

<p style='text-indent:20px;'>One of the challenges facing supply chain designers is designing a sustainable and resilient supply chain network. The present study considers a closed-loop supply chain by taking into account sustainability, resilience, robustness, and risk aversion for the first time. The study suggests a two-stage mixed-integer linear programming model for the problem. Further, the robust counterpart model is used to handle uncertainties. Furthermore, conditional value at risk criterion in the model is considered in order to create real-life conditions. The sustainability goals addressed in the present study include minimizing the costs, <inline-formula><tex-math id="M1">\begin{document}$ \text{CO}_2 $\end{document}</tex-math></inline-formula> emission, and energy, along with maximizing employment. In addition, effective environmental and social life-cycle evaluations are provided to assess the associated effects of the model on society, environment, and energy consumption. The model aims to answer the questions regarding the establishment of facilities and amount of transported goods between facilities. The model is implemented in a car assembler company in Iran. Based on the results, several managerial insights are offered to the decision-makers. Due to the complexity of the problem, a constraint relaxation is applied to produce quality upper and lower bounds in medium and large-scale models. Moreover, the LP-Metric method is used to merge the objectives to attain an optimal solution. The results revealed that the robust counterpart provides a better estimation of the total cost, pollution, energy consumption, and employment level compared to the basic model.

Assessment of financial risk in the design and scheduling of multipurpose plants under demand uncertainty

ABSTRACT Industrial companies are seeking for highly flexible strategic and operational solutions to face the requirements of current dynamic markets. The aim of this work is to provide a decision support assessment for the design and scheduling of a multipurpose plant under demand uncertainty, allowing the assessment of alternative risk profile solutions. A general two-stage mixed-integer linear programming (MILP) model is proposed with the goal to maximise the annualised profit of the plant operation under a set of scenarios while minimising the associated financial risk. Considering the long-term investment perspective, the Conditional Value at Risk (CVaR) measure is used to evaluate the likelihood that a specific loss or gain will exceed a certain value at risk. A bi-objective model is formulated using the augmented ε-constraint method to generate an approximation to the Pareto-optimal curve, illustrating the trade-offs between plant profit (with the corresponding design and scheduling decisions) and the associated financial risk. Addressing a set of propositions regarding a case-study, the conclusions highlight the advantages of the risk measure integration in support of the decision-making process, discussing the managerial insights in the assessment of diverse financial outcomes for the solution optimisation.

A Bender’s based nested decomposition algorithm to solve a stochastic inland waterway port management problem considering perishable product

The inland waterway transportation system provides one of the most economical and environmentally friendly means of transportation with significant contributions to the nation’s overall transportation economy. This study aims to develop a sound and realistic model, capturing diversified inland waterway transportation network-related properties and complex interactions between different transportation entities. Additionally, this study ensures optimal inventory management decisions for perishable products having stochastic availability under unpredictable waterway conditions over time. To this end, we propose a two-stage mixed-integer linear programming (MILP) model capturing the aforementioned issues, along with specific concern to the perishable product storage and transportation. Subsequently, we propose a hybrid decomposition algorithm combining the enhanced Benders decomposition algorithm and sample average approximation to solve the large size test instances of this complex problem. Further, a case study considering the inland waterway transportation system of the lower Mississippi River is demonstrated. The sensitivity analysis results show that the system is highly sensitive to the commodity shelf life. With a 60% higher commodity deterioration rate, the overall commodity storage need increases by 12.3%, the total system cost increases by about 33%.

Flexibility enhancement in active distribution networks through a risk-based optimal placement of sectionalizing switches

The advent of distributed energy resources (DERs) in distribution networks, referred to as active distribution networks (ADNs), brings in new opportunities for distribution network operators (DNOs) to improve the network reliability. The contribution of DERs in reliability enhancement generally depends on the island mode operation of DERs. Optimal placement of sectionalizing switches (SSs) enables ADNs to be operated more reliably by islanding the faulted part of the system in the form of flexible micro-grids (FMGs). However, practical methods are needed to handle the complex problem of the SS placement in ADNs. This paper presents a risk-based two-stage mixed-integer linear programming (MILP) model for the optimal placement of the network-driven flexibility providers, i.e. SSs, in ADNs taking into account the limits of the resource-driven flexibility providers, i.e. DERs. The performance of the proposed model is evaluated on various test distribution networks, and its effectiveness demonstrated through several case studies.

A graph theory approach for scenario aggregation for stochastic optimisation

The development of fast, robust and reliable computational tools capable of addressing process management under uncertain conditions is an active topic in the current literature, and more precisely for the process systems engineering one. Particularly, scenario reduction strategies have emerged as an alternative to overcome the traditional issues associated with large-scale scenario-based problems. This work proposes a novel and flexible scenario-reduction alternative that integrates data mining, graph theory and community detection concepts to represent the uncertain information as a network and identify the most efficient communities/clusters. The capabilities of the proposed approach were tested by solving a set of two-stage mixed-integer linear programming problems under uncertainty. For comparison and validation purposes, these problems were also solved using two available methods (SCENRED and OSCAR). This comparison demonstrates that the results obtained by using the proposed approach are at least as good or better, in terms of quality and accuracy, than the results obtained bu using SCENRED and OSCAR. Additionally, the practical advantage of the proposed parameter definition rule is demonstrated as a way to overcome the limitations of the current alternatives (i.e. arbitrary user-defined parameters).

Beijing subsidiary administrative center multi-energy systems: An optimal configuration planning

Multi-energy systems (MES) contribute to increasing energy utilization efficiency and renewable energy accommodation by coupling multiple energy sectors. Beijing is planning to build a subsidiary administrative center in Tongzhou District. A new MES will be built in this center from scratch to jointly meet electricity, heat and cooling demands. This raises the need for optimizing the configuration of MES from scratch at the planning stage. In this paper, the configuration planning of the MES in Beijing’s new subsidiary administrative center is conducted using a two-stage mixed-integer linear programming (MILP) approach based on the energy hub (EH) model. Given the load demand, distributed renewable energy sources, energy prices and candidate system component parameters, the MES configuration planning, equipment selection and capacity planning are jointly managed without limiting the possibility of MES configuration. A sensitivity analysis is performed to show the impacts of load profiles and energy prices on the optimal MES configuration. Operation simulations are carried out based on the obtained optimal planning scheme and another four alternative planning schemes presented by Beijing subsidiary administrative center. The operating cost, CO2 emission and overall system efficiency of each planning scheme are calculated and compared with each other.

A data-driven hierarchical MILP approach for scheduling clinical pathways: a real-world case study from a German university hospital

Facing economic pressure and case-based compensation systems, hospitals strive for effectively planning patient hospitalization and making efficient use of their resources. To support this endeavor, this paper proposes a flexible hierarchical mixed-integer linear programming (MILP)-based approach for the day-level scheduling of clinical pathways (CP). CP form sequences of ward stays and treatments to be performed during a patient’s hospitalization under consideration of all relevant resources such as beds, operating rooms and clinical staff. Since in most hospitals CP-related information needed for planning is not readily available, we propose a data-driven approach in which the structure of the CP to be scheduled including all CP-related constraints is automatically extracted from standardized hospital billing data available in every German hospital. The approach uses a flexible multi-criteria objective function considering several patient- and hospital-related aspects which makes our approach applicable in various scenarios. Furthermore, in contrast to other approaches, it considers several practically relevant aspects ensuring the implementability of the scheduling results such as multiple ward stays per hospitalization and gender-separated room assignments. Regarding the treatment resources such as operation rooms and clinical staff, it considers the eligibility of resources for treatments based on information such as special equipment or qualification and represents complex resources individually to avoid disaggregation problems. To allow solving the resulting complex and large-scale scheduling problem for realistically dimensioned problem instances, we propose a hierarchical two-stage MILP approach involving carefully designed anticipation components in the first-stage model. We evaluate our approach in a case study with real-world data from a German university hospital showing that our approach is able to solve instances with a planning horizon of 1 month exhibiting 1088 treatments and 302 ward stays of 286 patients. In addition to comparing our approach to a monolithic MILP approach, we provide a detailed discussion of the scheduling results for two practically motivated scenarios.