Mixed Integer Linear Programming Research Articles

Trolley Optimisation for Loading Printed Circuit Board Components

A trolley is a container for loading printed circuit board (PCB) components, and a trolley optimisation problem (TOP) is an assignment of PCB components to trolleys for use in the production of a set of PCBs in an assembly line. In this paper, we introduce the TOP, a novel operation research application. To formulate the TOP, we derive a novel extension of the bin packing problem. We exploit the problem structure to decompose the TOP into two smaller, identical, and independent problems. Further, we develop a mixed integer linear programming model to solve the TOP and prove that the TOP is an NP-complete problem. A case study of an aerospace manufacturing company is used to illustrate the TOP which successfully automated the manual process in the company and resulted in significant cost reductions and flexibility in the building process.

Customised bus service design considering flexible vehicle size and transfer incentivization

In this study, the flexible vehicle size and transfer incentivization strategies are incorporated to increase the potential of the customised bus system. In detail, the modular vehicle system is adopted for flexible vehicle size design since it can be connected as the assembled bus in response to demand variation. Also, the transfer among buses is considered to group passengers with the same destinations to increase bus utilisation. However, the passengers will not actively transfer as it is viewed as a disutility. Thus, the incentivization is provided and the transfer demand is elastic under incentivization. We jointly optimise passenger-route assignment, vehicle size, and incentivization scheme by transforming the original nonlinear programming model into a mixed integer linear programming model. Then, to solve the larger case study, the modified ant colony system algorithm is also proposed. The experiments validate the superiority of customised bus service considering flexible vehicle size and transfer incentivization.

A topology and sizing optimisation method for lightweight sandwich structures subject to dynamic and static constraints

A static-dynamic topology-sizing optimisation method is presented. The solution is based on a sequential Mixed-Integer Linear Programming solution and aims to minimise the mass of a structure subjected to concurrent constraints on static and dynamic response. It is shown that the classical problem of the dynamics of lightweight sandwich structures may be mitigated through core topology and face sheet thickness combinations, retaining the static load carrying capacity while presenting stringent dynamic properties at a low mass penalty.A numerical example, in the form of a load carrying sandwich beam which is excited at different frequencies, is used to demonstrate the method.

Designing a centralized storage hydrogen supply chain network with multi-period and bi-objective optimization

This study introduces a multi-period centralized storage optimization model aimed at designing an efficient hydrogen supply chain system, considering cost and emissions as dual objectives. It integrates multiple energy sources, production and storage methods, transport combinations, demand scenarios, and carbon capture systems, offering a comprehensive decision-making approach for hydrogen network design. Employing the mixed-integer linear programming methodology, the proposed model resolves these complexities. The research applies this model to a case study in France, generating six unique scenarios for 10 and 15 cities, and compares them against two distinct decentralized models. The findings consistently highlight the centralized storage model’s cost benefits across various demand scenarios, including cases of unrestricted emissions as well as cases with limited emission targets. The cost-effectiveness of this proposed model enhances its feasibility within the current context of decarbonization.

Integrated food delivery problem considering both single-order and multiple order deliveries

The global food delivery market is rapidly growing, and leading companies in Korea have recently introduced a single-order delivery service, wherein a courier picks up customers’ food and delivers it immediately without visiting other places. Although this delivery service is associated with a high delivery fee, it is popular among customers who prefer fresh and fast delivery, unlike multiple-order delivery services, which handle orders from several customers simultaneously. Nevertheless, the coexistence of both delivery services has led couriers to prefer single-order deliveries because they are more profitable, resulting in intensified competition among couriers. Furthermore, there are cases where the courier visits other restaurants or customers without immediately delivering food to customers who use the single-order delivery service, making it a multiple-order delivery service, which leads to customer dissatisfaction. To address these issues, this study proposes a mixed-integer linear programming model for optimizing order assignment and routing in an environment where both single-order and multiple-order deliveries coexist. The tabu search was used to solve medium-sized and large-sized problems, and several food delivery models were configured by referencing the operating methods of the platforms. Through experiments, we demonstrate that the proposed model is more efficient in properly delivering food to customers for both single and multiple orders than other food delivery models. We also analyzed the impact of various variables through a sensitivity analysis, providing insights into the platform.

Optimal Network Design for Municipal Waste Management: Application to the Metropolitan City of Rome

This work is part of the regional research project PIPER—Intelligent Platform for the Optimization of Recycling Operations, which aims to develop a network design model for waste collection in the metropolitan city of Rome, Italy. The goal is to enhance the strategic planning of treatment and disposal facilities by incorporating transshipment locations to improve the efficiency of waste collection. The motivation for this study stems from the stringent targets set by the European Green Deal to achieve near-zero emissions and the critical role of waste management in realizing circular economy practices. Methods: The problem is formulated as a mixed integer linear program (MILP) that includes constraints on vehicle allocation to shifts and driver requirements. An additional feature of the model is its ability to account for vehicle maintenance issues when deployed on consecutive shifts. Results: The model was tested against a real-world case study in Rome and demonstrated significant potential savings, with a reduction in total traveled distance exceeding 10%. The mathematical model was also used for evaluating strategic scenarios in a “what-if” analysis, allowing the municipal collection company to assess location options for depots, waste treatment facilities, and transshipment points, as well as to optimize the fleet composition and driver allocation during work shifts. Conclusions: This study presents a robust tool for strategic planning in waste collection operations, highlighting the benefits of using transshipment locations to improve efficiency. The findings indicate substantial potential for cost savings and operational improvements, making the model a valuable asset for municipal waste management planning.

Stochastic energy planning of a deltoid structure of interconnected multilateral grids by considering hydrogen station and demand response programs

This paper presents triple-objective stochastic energy planning and management of a deltoid structure in which a microgrid, nano-grid, and main grid connect and exchange power simultaneously. In addition, the impact of hydrogen stations due to the growth of hydrogen vehicles and their crucial role in the power system's future to reduce pollution is also discussed. Moreover, the effect of time-based demand response programs (TOU) according to the elasticity matrix (different operators and price-sensitive flexible loads) for the proposed multilateral grid is investigated under diverse scenarios. Stochastic planning is performed to make results more realistic and authentic. The uncertain parameters for stochastic planning include wind pace, solar radiation, fuel rate, and various demands. The assumed triple objective functions for the proposed planning are the microgrid's profit, the nano-grid's cost, and the total multilateral grid's pollution. The problem is modeled as mixed-integer linear programming (MILP) and solved using the GAMS and LP metric approach. The final results show that by implementing the supposed planning, the microgrid's profit increases by about 22.53 $/day (10.8%), and the nano-grid's cost decreases by about 1.31 $/day (9.8%). On the other hand, the total environmental pollution is reduced significantly and reaches 1.06 kg/day.

Learning to Allocate Time-Bound and Dynamic Tasks to Multiple Robots Using Covariant Attention Neural Networks

Abstract In various applications of multi-robotics in disaster response, warehouse management, and manufacturing, tasks that are known a priori and tasks added during run time need to be assigned efficiently and without conflicts to robots in the team. This multi-robot task allocation (MRTA) process presents itself as a combinatorial optimization (CO) problem that is usually challenging to be solved in meaningful timescales using typical (mixed)integer (non)linear programming tools. Building on a growing body of work in using graph reinforcement learning to learn search heuristics for such complex CO problems, this paper presents a new graph neural network architecture called the covariant attention mechanism (CAM). CAM can not only generalize but also scale to larger problems than that encountered in training, and handle dynamic tasks. This architecture combines the concept of covariant compositional networks used here to embed the local structures in task graphs, with a context module that encodes the robots’ states. The encoded information is passed onto a decoder designed using multi-head attention mechanism. When applied to a class of MRTA problems with time deadlines, robot ferry range constraints, and multi-trip settings, CAM surpasses a state-of-the-art graph learning approach based on the attention mechanism, as well as a feasible random-walk baseline across various generalizability and scalability tests. Performance of CAM is also found to be at par with a high-performing non-learning baseline called BiG-MRTA, while noting up to a 70-fold improvement in decision-making efficiency over this baseline.

Scheduling in a Simple Assembly Robotics Cell to Minimize Earliness and Tardiness

Background: Robotics Assembly Cells (RAC) have been designed to meet the flexibility requirements demanded by today's globalized market. The objective is to manufacture a vast variety of products at a low cost, which requires equipment with a high level of flexibility, such as robots. The need to schedule a great variety of jobs in an RAC is a very relevant issue, as efficiency and productivity depend on the sequence in which jobs are scheduled. Studies around this matter have developed models with analytical and heuristic approaches, as well as simulation methods and genetic algorithms, seeking to improve performance measures based mainly on time, utilization, and costs. Method: The purpose of this article is to formulate an exact mathematical model using mixed-integer linear programming (MILP) to optimize small scheduling problems. The objective is to minimize the measure of performance related to the tardiness and earliness of jobs. This optimization aims to mitigate the effects of delays in product deliveries, queue times, and work-in-process inventory in subsequent processes. Doing so facilitates adherence to agreed-upon delivery deadlines and prevents bottlenecks in the assembly cell. Results: The proposed mathematical model generates optimal solutions to the job scheduling problem in the assembly cell, which serves as a case study. This addresses the need to minimize tardiness to meet delivery deadlines or minimize earliness while avoiding an increase in work-in-process inventories. The model ensures that optimal scheduling decisions are made to optimize both delivery performance and inventory levels. Conclusions: Due to the NP-hard complexity of the scheduling problem under study, the proposed mathematical model demonstrates computational efficiency in solving scheduling problems with fewer than 20 jobs. The model is designed to handle such smaller-scale problems within a reasonable computational time frame, considering the inherent complexity of the scheduling problem.

Integral techno-economic design & operational optimization for district heating networks with a Mixed Integer Linear Programming strategy

The Netherlands, with over 90% of homes heated by natural gas is currently in the early phases of the heat transition to find alternative solutions towards 2050. According to ambitions of the Dutch government up to 50% of future implemented heating systems in the built environment will use District Heat Networks. The investment-heavy nature for District Heating System (DHS) makes it challenging to establish viable business-cases as supply-side parties require security of demand. However, resident participation is lacking as there is no integral estimate for the cost of heat over the lifetime in the early planning phase. This paper proposes a network integral techno-economic optimization with minimal a-priori assumptions. An integral network cost optimization enables to achieve a considerably more reliable cost of heat in the early planning phase. Both the investments and operational strategy are optimized with a Mixed Integer Linear Programming approach that captures the physics as well as the financial choices. Linearization of the physics are chosen to have a conservative impact on the costs estimates. The end scenario network is designed where the placement and size of sources, storage components and pipes are optimized together with the operational strategy, e.g. thermal allocation time-series, for all assets. The workflow was applied to a greenfield network for the Dutch municipality of Rijswijk. It was shown that a 18% reduction in Total Cost of Ownership in the primary grid could be achieved by introducing decentralized sources, decentralized storages and seasonal storage.

Creating a renewable energy-powered energy system: Extreme scenarios and novel solutions for large-scale renewable power integration

The large-scale integration of renewable power sources, such as wind and solar, is pivotal in achieving climate change mitigation goals and reducing dependency on fossil fuels within energy systems. This manuscript explores the integration of substantial wind and solar power capacities into Luxembourg's energy framework, addressing challenges like variability, intermittency, and curtailment. It delves into scenarios that include power-to-heat (P2H) and vertical farming (VF) technologies to improve system flexibility and promote the incorporation of renewable energy. Utilising a mixed-integer linear programming (MILP) algorithm, an energy balance model with hourly resolution optimises the system. This algorithm guides the operation of individual plants, ensuring optimal performance based on a variety of criteria. Notable findings underscore the significant influence of an appropriate mix of renewable power resources, indicating a potential increase in renewable power consumption by up to 50% within the system. Furthermore, curtailment management solutions are shown to enhance renewable power integration by as much as 30%, highlighting the importance of strategic, system-wide approaches. Additionally, vertical farming emerges as a promising avenue for significantly augmenting the utilisation of surplus solar power, potentially by up to 100%. While the integration of large-scale renewable power is demonstrably viable, it necessitates a meticulously crafted strategy that considers technological innovations and strategic planning.

Reliability enhancement through optimal placement of photovoltaic power plant and battery energy storage in distribution system

Integration of distributed generation (DG) units could reduce the duration of power outage for a certain number of consumers in distribution networks after a network failure. Prerequisites are that i) the DG unit has the technical characteristics that enable its islanded operation, the degree of automation of the distribution network allows it and ii) it is in accordance with the current technical regulations of the country. The extent to which integration of distributed sources can improve reliability depends on the DG size, type, and the point of common coupling. Besides, an energy storage system could be installed along with DG unit, so that energy supply availability during fault periods can be secured. This paper proposes an algorithm based on mixed-integer linear programming (MILP) approach for optimal placement of photovoltaic power plant (PVPP) and battery energy storage system (BESS). The optimal location is determined so that expected non supplied energy is minimized. The uncertainties in the estimation of production of PVPP, load and BESS state of charge (SoC) were taken into account by Monte Carlo simulations (MCS).

Low‐carbon economic schedule of the H2DRI‐EAF steel plant integrated with a power‐to‐hydrogen system driven by blue hydrogen and green hydrogen

AbstractHydrogen direct reduction iron coupled with electronic arc furnace (H2DRI‐EAF) technology, as an important technology for decarbonisation in the iron and steel industry, has the advantages of high electrification and low carbon emissions. However, the large demand for hydrogen in this technology relies significantly on the production of electrolytic hydrogen, leading to a substantial increase in power consumption in the steel production process. Moreover, the use of an unclean power source in electrolytic hydrogen production leads to increases in indirect carbon emissions, reducing the low‐carbon attributes of the technology. This study investigates the integrated flexible operation mode of a steel plant. An illustrating method is utilised for modelling the entire steel production process and power to hydrogen (PtH2) process in detail for the H2DRI‐EAF steel plant, which includes natural gas, photovoltaic, wind power self‐provided power plants, and carbon capture and storage (CCS) systems. A mixed integer linear programming (MILP) model is developed for the comprehensive scheduling of the steel mill. The results of the case studies indicate that by reliably integrating the production of renewable energy and natural gas power plants, the PtH2 system can fully consume the renewable energy output while ensuring the smooth progress of steel production and maximising the reduction of carbon emissions from hydrogen production and the total cost of steel production.

Preemptive and non-preemptive multi-skill multi-mode resource-constrained project scheduling problems considering sustainability and energy consumption: A comprehensive mathematical model

Modern project managers cope with significant challenges to schedule and control projects considering dynamic environments, frequent uncertainties, strict project deadlines, and stricter sustainable requirements above all. Sustainability taking into account resource utilization has been recently associated with project management. Hence, this paper presents a new mixed-integer linear programming (MILP) model with two objectives for a resource-constrained project scheduling problem (RCPSP) with multiple skills and multiple modes, assuming preemptive and non-preemptive activities in an uncertain environment. Given the importance of sustainable developments in projects, the considered objectives are to maximize job opportunities and minimize project duration, resource costs, and total energy consumption. To deal with the model, an AUGNMECON2VIKOR algorithm is utilized to create Pareto solutions. In this model, project activities can be crashed by allocating extra resources. Furthermore, multi-skill resources are used to perform project activities. This study also investigates the impact of these resources on project scheduling. To deal with uncertain circumstances, a fuzzy chance-constrained programming method is employed to develop a robust possibilistic programming model. With respect to the increasing significance of sustainability in project management, this study pioneers the examination of the impact of sustainable factors on project scheduling. Finally, the proposed formulation is validated using instances from the well-known PSPLIB and MMLIB test sets. Finally, a comparison is drawn between the presented solution method considering AUGMECON2VIKOR and AUGMECON2.

The Efficacy of a Cellular Approach to Resilience-Oriented Operation of Distribution Grids Based on Eigenvectors of the Laplacian Matrix

This study addresses the critical question for distribution system operators: how to maintain power supply during an impending disaster that could disrupt the grid? To address this challenge, we propose a cost-efficient cellular model for enhancing the resilience of smart distribution grids. This model prioritizes resilience in the face of natural disasters or other disruptions that could impact service delivery. This approach benefits both grid operators and consumers by ensuring reliable power supply while minimizing energy costs. Furthermore, the model's scalability allows it to be applied to distribution grids of varying sizes. The proposed method utilizes an innovative approach to form optimal cellular network configurations within the grid. As the first step in the formation of cellular topology for the grid, the eigenvectors of the Laplacian matrix of the grid will be used to decide on the optimal configurations. Subsequently, a bi-level mixed-integer linear programming model is proposed to decrease the network costs while simultaneously consider potential power transfer scenarios between the cells and the upstream network during both normal and emergency conditions. The researchers validated the effectiveness of the proposed method through simulations on an IEEE 33-bus test system. The results demonstrate outstanding performance, with a significant increase in the resilience index (around 96%) and a substantial reduction in load-shedding costs (around 80%), making the network considerably more robust.

Energy management in integrated energy system with electric vehicles as mobile energy storage: An approach using bi-level deep reinforcement learning

The integrated energy system with electric vehicle charging station via vehicle-to-grid aims to offer a proactive solution for low-carbon development of both energy and transportation sectors. However, achieving optimal energy efficiency with minimal operational costs in such a complex system is challenging due to the high randomness of electric vehicle travel patterns. This work proposes a reinforcement learning-based energy management framework to optimize the coordinated scheduling of integrated energy and vehicle-to-grid. The problem is modeled as a Markov Decision Process and optimized using the bi-level soft actor-critic algorithm, where the upper-level agent supervises the scheduling of the integrated energy system, and lower-level agent sets the electric vehicles’ charging and discharging strategies based on upper-level decision. Numerical simulations demonstrated that by adopting a bi-level reinforcement learning approach, the proposed algorithm effectively enhances energy exchange between integrated energy and electric vehicle charging station, reducing operational costs by 8 % compared to other multi-agent algorithms. Furthermore, when compared to the Mixed-Integer Linear Programming model, the proposed algorithm achieves similar optimality with significantly less computation time, maintaining a margin of error of approximately 4 %. Additionally, integrating electric vehicles as mobile energy storage within this framework can lead to a further 10 % reduction in operating costs.

Designing a Sustainable Water Supply Strategy through Biobjective Mixed-Integer Linear Programming: A Case Study on Gaza

Designing a Sustainable Water Supply Strategy through Biobjective Mixed-Integer Linear Programming: A Case Study on Gaza

Minimising makespan in distributed assembly hybrid flowshop scheduling problems

To enhance the manufacturing flexibility, resilience, and production efficiency, the integration of scheduling for distributed manufacturing with assembly systems has become a pivotal driver of production planning evolution. In this research endeavour, we present a Mixed-Integer Linear Programming model and an innovative Iterated Epsilon-Greedy Reinforcement Learning algorithm to address the distributed assembly hybrid flowshop scheduling problem. Empirical validation, conducted through computational experiments on a benchmark problem set, is used to gain important managerial insights. The computational results demonstrate that the proposed algorithms significantly reduce the makespan for the addressed problem. This study has the potential to make valuable contributions to ongoing research endeavours within the realm of multi-stage shop scheduling, an area that continues to warrant progressive advancement.

An interactive framework for strategic participation of a price-maker energy hub in the local gas and power markets based on the MPEC method

Power-to-gas (P2G) technologies within smart energy hubs (SEH) offer unprecedented operational flexibility to distribution networks. This is achieved by integrating various energy carriers both regionally and locally, particularly in the presence of renewable energy sources (RES). According to this subject, the present work outlines an interactive risk-based model and meticulous mathematical solution to justifiably incorporate the P2G units into the SEH to participate in the local electricity and natural gas markets strategically. As a price-maker prosumer, the SEH utilizes cross-product arbitrage strategies to optimize its proposals for power and gas exchanges in the local markets. The participation model for SEH is developed as a bi-level framework. At the top level, the goal is to maximize SEH's profit while accounting for the risks posed by RES. At the bottom level, the aim is to increase social welfare under network constraints through local market settlements. Finally, a mathematical program with equilibrium constraints (MPEC) is offered as a solution to solve the bi-level model. Using integer disjunctions on the complementarity and slackness criteria and the strong duality theory on the bilinear product terms, a mixed-integer linear programming (MILP) has been derived to approximate the suggested MPEC formulation. It can be concluded from the results that the integration of P2G technologies and the use of mutual product arbitrage in the local electricity and natural gas markets, not only increases the profit of the SEH, but also reduces operating costs and improves social welfare as well as cost-recovery of the entire network.

Optimal Bidding Scheduling of Virtual Power Plants Using a Dual-MILP (Mixed-Integer Linear Programming) Approach under a Real-Time Energy Market

In recent years, the energy industry has increased the proportion of renewable energy sources, which are sustainable and carbon-free. However, the increase in renewable energy sources has led to grid instability due to factors such as the intermittent power generation of renewable sources, forecasting inaccuracies, and the lack of metering for small-scale power sources. Various studies have been carried out to address these issues. Among these, research on Virtual Power Plants (VPP) has focused on integrating unmanaged renewable energy sources into a unified system to improve their visibility. This research is now being applied in the energy trading market. However, the purpose of VPP aggregators has been to maximize profits. As a result, they have not considered the impact on distribution networks and have bid all available distributed resources into the energy market. While this approach has increased the visibility of renewables, an additional method is needed to deal with the grid instability caused by the increase in renewables. Consequently, grid operators have tried to address these issues by diversifying the energy market. As regulatory method, they have introduced real-time energy markets, imbalance penalty fees, and limitations on the output of distributed energy resources (DERs), in addition to the existing day-ahead market. In response, this paper proposes an optimal scheduling method for VPP aggregators that adapts to the diversifying energy market and enhances the operational benefits of VPPs by using two Mixed-Integer Linear Programming (MILP) models. The validity of the proposed model and algorithm is verified through a case study analysis.