Multi-objective Mixed Integer Linear Programming Research Articles

Optimization of Pickup Vehicle Scheduling for Steel Logistics Park with Mixed Storage

Pickup vehicle scheduling in steel logistics parks is an important problem for determining the outbound efficiency of steel products. In a steel logistics park, each yard contains different types of steel products, which provides flexible yard selection for each pickup operation. In this case, the yard allocation and the loading sequence for each vehicle must be considered simultaneously in pickup vehicle scheduling, which greatly increases the scheduling complexity. To overcome this challenge, in this paper, we propose a pickup vehicle scheduling problem with mixed steel storage (PVSP-MSS) to optimize the makespan of pickup vehicles and the makespan of steel logistics parks simultaneously. The optimization problem is formulated as a multi-objective mixed-integer linear programming model, and an enhanced algorithm based on SPEA2 (ESPEA) is proposed to solve the problem with a high efficiency. In the ESPEA, a cooperative initialization strategy is firstly proposed to initialize the vehicle pickup sequence for each yard. Then, an insertion decoding method is designed to improve the scheduling efficiency, utilizing the idle time of a yard. Furthermore, local search technology based on critical paths is proposed for the ESPEA to improve the solution quality. Experiments are executed based on data collected from a real steel logistics park. The results confirm that the ESPEA can significantly reduce both the makespan of each pickup vehicle and the makespan of the steel logistics park.

Designing a reliable-sustainable supply chain network: adaptive m-objective ε-constraint method

AbstractIn the current era emphasizing sustainability and circularity, supply chain network design is a critical challenge for making reliable decisions. The optimization of facility location-allocation inventory problems (FLAIPs) holds the key to achieving dependable product delivery with reduced costs and carbon emissions. Despite the importance of these challenges, a substantial research gap exists regarding economic, reliability, and sustainability criteria for FLAIPs. This paper aims to fill this gap by introducing a multi-objective mixed-integer linear programming model, focusing on configuring a reliable sustainable supply chain network. The model addresses three key objectives: minimizing costs, minimizing emissions, and maximizing reliability. A notable contribution of this research lies in elaborating on five levels of a supply chain network catering to the delivery of multiple products across various periods. Another novelty is the simultaneous incorporation of economic, environmental, and reliability objectives in the network design—a facet rarely addressed in prior research. Results highlight that varying demand levels for each facility lead to altered trade-offs between objectives, empowering practitioners to make diverse decisions in facility location allocation. The proposed mathematical model undergoes validation through numerical examples and sensitivity analysis of parameters. The paper concludes by presenting theoretical and managerial implications, contributing valuable insights to the field of sustainable supply chains.

An integrated Bi-objective green vehicle routing and partial disassembly line problem for electronic waste: an industrial case study

ABSTRACT Today, serious environmental problems arise with overconsumption. The most effective way of protecting the environment is recycling. To obtain maximum benefit from the recycling activities depends on the effective and robust design of the disassembly lines. In addition, the distribution plan of products to be recycled is as important as the disassembly process. Therefore, this study includes the integrated optimization of the partial disassembly line balancing problem and the multi-objective green vehicle routing problem for the first time. The integrated problem is formulated as a Multi-Objective Mixed Integer Linear Programming. The model’s objectives consist of minimization of total CO2 emission and cost. The Pareto optimal solution set is found by solving this problem with the Augmented ℇ-Constraint Method. The model respectively determines (i) the total number of opened workstations in the disassembly center, (ii) optimal task assignments to workstations in disassembly centers and non-disassembled tasks, (iii) total workstation times and idle times of each workstation, (iv) optimal distribution routes for all vehicles. This study examines a case study in Turkey. To determine the best Pareto optimal solution for homogeneous and heterogeneous fleets, the integrated multi-criteria decision-making method consisting of EWM (Entropy Weight Method) and COPRAS (Complex Proportional Assessment) is used.

A Socio-Economic and Quality-Oriented Optimal Fruit Supply Chain Network Design in a Multi-Market Multi-Product Environment: A Real Case Study

This paper focuses on developing an optimal sustainable fruit Closed-Loop Supply Chain (CLSC). A novel multi-objective Mixed-Integer Linear Programming (MILP) model is suggested to formulate a multi-product, multi-market, multi-period, multi-level, multi-mode selling problem which aims to minimize cost and emissions and maximize the whole social segments’ responsiveness to fruit, vinegar, and compost. The model is solved by the LP-metric method and implemented in a real case in Iran. The findings indicate that achieving the highest level of responsiveness comes with a substantial cost for the network. However, this does not apply to environmental considerations. For decision-makers prioritizing sustainability, the proposed decision support system strongly advises against buying fruit from fruit cars. Due to the absence of positive impacts on the level of responsiveness, buying from fruit cars results in increased costs and emissions. For a cost- and environmentally-sensitive decision-maker, it is recommended to provide higher-quality fruits. Because the lower the quality of the fruit, leads to increased waste rates and emissions. Meeting increased vinegar demand, with its elevated costs and emissions, becomes justifiable only when the policymaker is actively seeking to achieve maximum responsiveness.

Optimal sizing of behind-the-meter battery energy storage systems under optimal battery operation: A case study in Ireland

The global shift toward a sustainable future has led to new policies to promote the use of renewable energy resources (RESs) in residential buildings. In Ireland, the number of installed roof-top photovoltaics (PVs) and home batteries, known as behind-the-meter (BTM) resources, has significantly increased in recent years, accelerated by grants administered by the Sustainable Energy Authority of Ireland (SEAI). In this work, an optimization tool that uses a combination of a Genetic Algorithm (GA) and a multi-objective mixed-integer linear programming (MOMILP) model is developed. This tool integrates an optimal battery control scheme from the MOMILP model aiming to minimize electricity bills and battery degradation to optimize the sizing of BTM batteries with minimum installation costs. Various scenarios have been simulated to assess the potential of the proposed model under different metering and billing systems, and optimization constraints. Compared to a case with a rule-based optimization model, results show that our optimization model can save 5.3 % more on annual bill costs while keeping battery degradation to a minimum of 2.1 % per year.

Evaluation of disassembly line layouts using an integrated fermatean fuzzy decision-making methodology: An application for refrigerator disassembly line

The layout focusing on the physical arrangement of workstations is an important consideration when designing a disassembly line. The layout of a line has a significant impact on its efficiency. The new problem addressed within this study is evaluating the main disassembly line layout types with respect to a thorough hierarchy of evaluation criteria classifying quantitative and qualitative. The addressed problem is modeled as multi-criteria decision-making (MCDM). To this end, a novel methodology that combines Stepwise Weight Assessment Ratio Analysis (SWARA) and Weighted Aggregated Sum Product Assessment (WASPAS) methods under the Fermatean fuzzy environment is proposed for sustainable waste management. Fermatean fuzzy SWARA (FF-SWARA) is used to find out the weight of criteria consisting of disassembly line balancing characteristics, reconfigurability, working conditions, and end-of-life product characteristics. Fermatean fuzzy WASPAS (FF-WASPAS) is used to rank the alternative disassembly line layouts consisting of straight, U-shaped, two-sided, and parallel lines considering the weight of criteria. The multi-objective mixed integer linear programming (MILP) model for the disassembly line balancing problem (DLBP) of each line layout type is integrated into the proposed methodology. This integration reflects the effect of balancing performance measurements such as the number of stations, removing hazardous parts early, removing high demand parts early, and line lead time. Reconfigurability principles, working conditions, and product characteristics that rely on expert opinions are considered under the qualitative criteria. The developed methodology was applied to a refrigerator disassembly line and verified by sensitivity analysis and comparative studies. The results show that the two-sided disassembly line layout is the best option.

Redesign for flexibility through electrification: Multi-objective optimization of the operation of a multi-energy industrial steam network

Steam networks are attractive for contributing to the decarbonization of energy systems. The partial or complete electrification of steam generators is expected to reduce direct carbon emissions by replacing fossil fuels. Still, this solution poses challenges in the operation of industrial sites. Most studies on the optimization of steam networks focus on single-period problems that disregard the dynamic operation of flexibility and consider only costs or profit as their objective function. In this study, we analyze the effects of the electrification of steam generators in a multi-period, multi-objective problem, which accounts for profit from energy exchanges, direct and indirect emissions, and steam blow-off. To that end, we use data from an industrial steam network and model it as a multi-objective mixed integer linear program. Five different design scenarios based on flexibility measures in steam networks are formulated with data from an industrial case study. Each problem is solved iteratively using the AUGMENCON-R algorithm to obtain points in the Pareto front efficiently. Electrification together with the inherent storage capacity of the steam network resulted in a potential average profit increase of 488 EUR/day, a potential average emissions reduction of 17.8 tCO2/day, and a potential average blow-off steam reduction of 5.7 t/day. Furthermore, our multi-objective approach reveals an operational carbon emissions abatement cost ranging from 62.5 to 363.6 EUR/tCO2, depending on exogenous data.

Bi-objective optimization modeling for biomass supply chain planning

Biomass energy plays an essential role in renewable energy for many reasons, such as reducing the dependence on fossil fuels and lowering greenhouse gas emissions, providing heat, electricity, and biofuels for various applications, and utilizing waste materials for helpful energy products. Besides, it can create employment opportunities and promote rural development, especially in developing countries where biomass resources are abundant and accessible. In the context of renewable energy research and application, this paper aims to develop a multi-objective mixed integer linear programming for designing multiple echelon biomass supply chain networks. The model is formulated to consider the economic costs and environmental impact of biomass distribution from the suppliers to the biomass plants. In this research, the Epsilon constraint method is adopted to generate Pareto fonts, which provides the trade-offs between two objectives. Moreover, sensitivity analysis is implemented to provide decision-makers with information about a network with changed parameters such as demand. Our model allows the decision maker to determine the capacity of warehouses and biomass power plants, inventory levels, type of trucks, etc. The proposed model is verified and evaluated using a practical dataset from Can Tho province, Central Mekong River Delta in Vietnam, generating several benefits for energy security and sustainability. Such a network includes 3 types of power plants, 3 scales of warehouses, 13 potential locations, and 41 suppliers. From the generated solutions, with the proportion of biomass electricity satisfaction varying from 5% to 30%, Hung Phu, O Mon, and Cai Rang industrial parks are the most suitable for power plants.

Optimal sizing and energy management of a microgrid: A joint MILP approach for minimization of energy cost and carbon emission

The optimal design and operation of microgrids involves complex trade-offs between technical, economic, and environmental factors. This research addresses these challenges by proposing a comprehensive approach that combines the sizing and energy management problems of a microgrid into a single decision-making framework. The study introduces a joint multi-objective mixed-integer linear programming algorithm that minimizes the levelized cost of energy (LCOE) and life cycle emissions (LCE) of the microgrid formed by photovoltaic system, battery storage, and with/without grid connections. By considering 25-year project lifetime horizon and hourly energy management, the algorithm ensures an optimal solution that accounts for long-term planning and real-time operation. The research conducts an analysis of French electricity grid as a case study where a Pareto-front is created using the trade-off constraint approach. Furthermore, the impact of peak shaving is assessed on Pareto-front for various subscription power to utility grid. Lastly, the research examines energy sources in the French electricity grid, revealing insights into the energy mix’s implications for microgrid design. The results showed that LCOE and LCE can vary up to 28% and 17%, respectively, while the local renewable energy utilization can be increased to 54% by limiting the grid subscription power at 36 kVA.

An outer approximation algorithm for generating the Edgeworth–Pareto hull of multi-objective mixed-integer linear programming problems

AbstractIn this paper, we present an outer approximation algorithm for computing the Edgeworth–Pareto hull of multi-objective mixed-integer linear programming problems (MOMILPs). It produces the extreme points (i.e., the vertices) as well as the facets of the Edgeworth–Pareto hull. We note that these extreme points are the extreme supported non-dominated points of a MOMILP. We also show how to extend the concept of geometric duality for multi-objective linear programming problems to the Edgeworth–Pareto hull of MOMILPs and use this extension to develop the algorithm. The algorithm relies on a novel oracle that solves single-objective weighted-sum problems and we show that the required number of oracle calls is polynomial in the number of facets of the convex hull of the extreme supported non-dominated points in the case of MOMILPs. Thus, for MOMILPs for which the weighted-sum problem is solvable in polynomial time, the facets can be computed with incremental-polynomial delay—a result that was formerly only known for the computation of extreme supported non-dominated points. Our algorithm can be an attractive option to compute lower bound sets within multi-objective branch-and-bound algorithms for solving MOMILPs. This is for several reasons as (i) the algorithm starts from a trivial valid lower bound set then iteratively improves it, thus at any iteration of the algorithm a lower bound set is available; (ii) the algorithm also produces efficient solutions (i.e., solutions in the decision space); (iii) in any iteration of the algorithm, a relaxation of the MOMILP can be solved, and the obtained points and facets still provide a valid lower bound set. Moreover, for the special case of multi-objective linear programming problems, the algorithm solves the problem to global optimality. A computational study on a set of benchmark instances from the literature is provided.

A Hybrid Heuristic-Exact Optimization for Large-Scale Home Health Care Problem.

During the COVID-19 pandemic, numerous people experiencing illness or senescence choose to receive home health care (HHC) services. However, a rapid increase in patients makes it a challenge to reasonably allocate nurses to provide HHC services under the condition of a paucity of nurse resources and patient time window constraints. To solve the large-scale HHC problem, a hybrid heuristic-exact optimization algorithm is proposed with three novel contributions. Firstly, a framework of hybrid heuristic-exact optimization is designed to solve the large-scale problem where a reasonable solution is difficult to obtain under constraints. Secondly, a multi-objective mixed-integer linear programming modelization is formulated to get a more diverse nurse assignment. Finally, an improved branch and bound algorithm is proposed to speed up computation for the large-scale problem. Computational results on different HHC instances from 25 to 1000 patients demonstrate that the proposed algorithm can optimize the HHC problem with more than 100 patients and can provide various assignments for different numbers of nurses, which the common algorithm cannot optimize.

Lead-time-based freight routing in multi-modal networks considering the Physical Internet

This paper addresses the problem of optimizing the transport of goods in the Physical Internet (PI) framework in a multi-modal setting using a multi-objective mixed-integer linear programming (MILP) approach. The model is specifically designed to meet the requirements related to modular shipments and PI-hubs, and in particular, determines the allocation of modular shipments to each transport mode in an intermodal setting. In doing so, parallel direct connection via road, the delivery times and the transportation costs are minimized. The model is applied to a numerical case study, to test its effectiveness to enhance freight transport efficiency within the PI framework, by exploiting, in particular, all the capacities of the available vehicles. In addition, a sensitivity analysis is conducted on some model parameters, to test its reaction to changes in the supply system and in the objective priorities. Results show that all the shipments are effectively transported between the origin and the destination terminals, they are divided into modules when necessary, and the selected transport modes, allocation strategy, and delivery times vary accordingly to the objective priorities.

Data-driven evacuation and rescue traffic optimization with rescue contraflow control

In response to local sudden disasters, e.g., high-rise office or residential building fire disasters, road occupation can cause conflicts, and traffic directions may be opposite between evacuation vehicles and rescue vehicles; moreover, lane contraflow can be adopted to meet these surge traffic demands. However, lane contraflow that provides more roads for rescue vehicles reduces the traffic supply in the evacuation direction. It is unclear how to control the number of contraflow roads used by rescue vehicles to coordinate evacuation and rescue traffic operations. Here, we adjust the critical rescue traffic volume of reversing the normal road traffic direction to control rescue contraflow. Additionally, we propose a multiobjective mixed integer linear programming formulation for evacuation and rescue traffic optimization. Additionally, considering that the upper limit of the critical rescue traffic volume is unknown and that the proposed formulation includes multiple objectives and multi-priority vehicle classes, a three-stage solving algorithm is developed. Next, a large-scale evacuation and rescue traffic optimization result dataset is obtained for the Nguyen–Dupuis road network, and the impact of different rescue contraflow control plans on evacuation and rescue traffic is studied based on data-driven statistical analysis. The results show that by adjusting the optimal rescue traffic route, the critical rescue traffic volume for reversing the normal road traffic direction can reduce the interference of rescue traffic to evacuation traffic operation performance without reducing rescue traffic operation performance, and can be used to coordinate evacuation and rescue traffic operation under rescue contraflow.

Handling Non-Linearities in Modelling the Optimal Design and Operation of a Multi-Energy System

Multi-energy systems (MESs) combining different energy carriers like electricity and heat allow for more efficient and sustainable energy solutions. However, optimizing the design and operation of MESs is challenging due to non-linearities in the mathematical models used, especially the performance curves of technologies like combined heat and power units. Unlike similar work from the literature, this paper proposes an improved piecewise linearization method to efficiently handle the non-linearities, models an MES as a multi-objective mixed-integer linear program (MILP), and solves the optimization problem over a year with hourly resolution to enable detailed operation and faithful system design. The method uses fewer linear pieces to approximate non-linear functions compared to a standard technique, resulting in lower complexity while preserving accuracy. The MES design and operation problem maximizes cost reduction and the rate of renewable energy sources. A case study on an MES with electricity and heat over one year with hourly resolution demonstrates the effectiveness of the new method. It allows for solving a long-term MES optimization problem in reasonable computation times.

Economic and environmental optimisation of mixed plastic waste supply chains in Northern Italy comparing incineration and pyrolysis technologies

In the quest for sustainable plastic waste management, understanding economic and environmental implications enables optimal selection of treatment technologies. This study presents a multi-objective mixed integer linear programming framework to optimise the supply chain for mixed plastic waste in Northern Italy. Two technologies are considered: incineration and pyrolysis. Results offer quantitative insights into economic and environmental performance, balancing trade-offs between maximising gross profit and minimising greenhouse gas (GHG) emissions. Economic optimisation favours incineration for treating mixed plastic waste, resulting in the highest gross profit of 115 M€ per year, and the highest net GHG emissions of about 680 kt CO2eq per year. When the aim is environmental optimisation, pyrolysis is preferred due to its lower GHG emissions of 387 kt of CO2eq per year and yielding a gross profit of 54 M€ per year. Trade-off Pareto optimal solutions were analysed to identify reasonable trade-off configurations between the two objectives.

Adaptive Multi-Objective Algorithm for the Sustainable Electric Vehicle Routing Problem in Medical Waste Management

This paper addresses the complex issue of managing medical waste transportation using electric vehicles, with the goal of minimizing both energy consumption and the risks associated with hazardous waste. A multi-objective mixed-integer linear programming model is introduced, incorporating practical factors such as time windows, partial recharge policy, load-dependent discharge, infection risk, and trips to waste disposal facilities. Our proposed method, a combination of the multi-objective evolutionary algorithm using decomposition (MOEA/D) with adaptive large neighborhood search (ALNS) and local search (LS) techniques, is referred to as MOEA/D-ALNS. This method demonstrates superior performance compared with the non-dominated sorting genetic algorithm, NSGA-II, modified MOEA/D and MOEA/D-LNS in benchmark instances with realistic assumptions. Our experimental results revealed an inverse correlation between energy consumption and risk objectives. Sensitivity analyses showed that eliminating time-window constraints results in more energy-efficient and safer routes while maintaining a slightly lower battery energy level can strike an ideal balance between energy consumption, risk, and battery health. This research contributes to the understanding of infectious medical waste management with its consideration of electric vehicles and waste disposal. It lays a solid foundation for future studies aiming to improve the sustainability and efficiency of medical waste routing practices.

Optimizing the COVID-19 cold chain vaccine distribution network with medical waste management: A robust optimization approach.

This paper investigates the distribution problem of the COVID-19 vaccine at the provincial level in Turkey and the management of medical waste, considering the cold chain requirements and the perishable nature of vaccines. In this context, a novel multi-period multi-objective mixed-integer linear programming model is initially presented over a 12-month planning horizon for solving the deterministic distribution problem. The model includes newly structured constraints due to the feature of COVID-19 vaccines, which must be administered in two doses at specified intervals. Then, the presented model is tested for the province of Izmir with deterministic data, and the results show that the demand can be satisfied and community immunity can be achieved in the specified planning horizon. Moreover, for the first time, a robust model is created using polyhedral uncertainty sets to manage uncertainties related to supply and demand quantities, storage capacity, and deterioration rate, and it has been analyzed under different uncertainty levels. Accordingly, as the level of uncertainty increases, the percentage of meeting the demand gradually decreases. It is observed that the biggest effect here is the uncertainty in supply, and in the worst case, approximately 30% of the demand cannot be met.

Fostering supply chain resilience for omni-channel retailers: A two-phase approach for supplier selection and demand allocation under disruption risks

This study aims to optimize supplier selection and demand allocation decisions for omni-channel (OC) retailers to achieve supply chain resilience under the potential disruption risks. A two-phase approach with resilience factors that covers three main sourcing issues (i.e., supplier evaluation, supplier selection, and demand allocation) is proposed to support the decision-making. In the first phase, we construct a five-dimensional evaluation framework for OC retailers to identify supplier preferences and a hybrid model that combines the best–worst method to determine the weights of the evaluation criteria and evidential reasoning to evaluate potential suppliers. In the second phase, the preferences obtained from multiple suppliers are integrated into a multi-objective mixed-integer linear programming model aiming to minimize expected cost and maximize total purchasing value and geographical segregation based on three key resilience strategies of multiple sourcing, geographic diversification, and local sourcing. The efficiency of the aforementioned resilience strategies as well as the solvability of the proposed model are then validated numerically using a real-world case study and various MOEAs. The outcomes could be used as a decision-making tool to assist OC retailers in the performance assessment and optimal demand allocation among the alternative suppliers by considering costs, purchase value, and resilience simultaneously.

A scenario-based robust possibilistic-flexible programming model for responsive supply chain network design with a performance-oriented solution methodology

This study addresses an integrated multi-objective responsive supply chain network optimization problem under hybrid uncertain business environment and optimal scheme selection. In addition to economic performance, delivery time and delivery reliability, as responsiveness indicators, are introduced into the multi-objective mixed-integer linear programming model. To deal with hybrid uncertainties caused by various scenarios, parameters, and flexible constraints, a scenario-based robust possibilistic flexible programming approach is extended. Moreover, a performance-oriented solution method is developed to solve the multi-objective optimization problem. The method integrates meta-heuristic and entropy-weighted TOPSIS to perform performance evaluation together with iteration optimization. The set of non-dominant solutions obtained by evaluation guide the iteration process. The numerical case verifies the effectiveness and efficiency of the proposed model, uncertainty technical and solution method. Interestingly, Pareto optimal solution set and its sort obtained by evaluation provide the optimal network scheme with the best supply chain performance.

Multi-objective optimisation of a carbon capture and sequestration supply chain under seismic risk constraints. A case study considering industrial emissions in Italy

Carbon capture and sequestration represents a key decarbonisation option, particularly for large-scale fossil-based industry sectors (e.g. cement, steel, and oil refineries). Carbon capture and sequestration networks have nonetheless raised concerns regarding the possibility of leakages, especially in high seismic-risk regions.In this study, a multi-objective mixed integer linear programming modelling framework is developed to minimise the cost and the seismic risk associated with the deployment of a carbon capture and sequestration infrastructure in Italy. The most significant industrial carbon dioxide sources (23 cement plants, 7 refineries, and 2 steel mills) are included in the model. The optimisation variables comprise capture technologies, transport options (onshore vs. offshore pipelines), and the choice of onshore and offshore deep saline aquifers for sequestration.Three carbon dioxide emission reduction targets (20 %, 50 %, 80 %) are considered to assess the optimal design configurations in terms of either cost or seismic risk. Results show that the seismic risk optimum determines an increase in total cost ranging between 10 % (for an 80 % reduction target) and 65 % (for a 20 % reduction target) with respect to the economic optimum. Considering only offshore sequestration leads to cost increase between 20 % and 30 % with respect to solutions accepting onshore sequestration, too. Conversely, cost-optimal infrastructures have a seismic risk that is between 1.5 and 18 times higher than that of the safest chains.