Mixed Integer Linear Model Research Articles

Approximating the Pareto frontier for bi-objective preventive maintenance and workshop scheduling: A Lagrangean lower bounding methodology for evaluating contracting forms

AbstractEffective planning of preventive maintenance plays an important role in maximizing the operational readiness of any industrial system. We consider an operating system and a maintenance workshop governed by two stakeholders who collaborate based on a mutual contract: components of the operating system that need maintenance are sent to the maintenance workshop, where necessary maintenance activities are performed and after which the maintained components are returned to the operating systems and ready to be used again. While the maintenance activities must obey the workshop capacity, the components should be returned to the operating system within a contracted time frame. For this problem, we developed in a previous work a mixed-integer linear optimization model incorporating stocks of damaged as well as repaired components, workshop scheduling, and preventive maintenance planning for the operating system. We then investigated an availability contract between the stakeholders and which is in the paper at hand compared with a turn-around time contract type, which is more often used in reality. Since, for real instance sizes, the latter leads to a computationally demanding bi-objective optimization problem, we use Lagrangean relaxation and subgradient optimization to compute local lower bounds on the set of non-dominated points, complemented with math-heuristics to identify good feasible solutions (i.e., local upper bounds). Our suggested method thus provides a bounding of the set of non-dominated points for a turn–around time contract.

A two-echelon multi-trip vehicle routing problem with synchronization for an integrated water- and land-based transportation system

This study focuses on two-echelon synchronized logistics problems in the context of integrated water- and land-based transportation (IWLT) systems. The aim is to meet the increasing demand in city logistics as a result of the growth in transport activities, including parcel delivery, food delivery, and waste collection. We propose two models, a novel mixed integer linear joint model, and a logic-based Benders’ decomposition (LBBD) model, for a two-echelon problem under realistic settings such as multi-trips, time windows, and synchronization at the satellites with no storage and limited resource capacities. The objective is to optimize transfers and satellite assignments, thereby reducing overall logistics costs for street vehicles and vessels. Computational experiments demonstrate that the LBBD model is more robust in terms of solution quality and solution time on average while the added value of the LBBD is more evident when solving large-scale instances with 100 customers, reducing the overall costs by 10.6% on average and significantly reducing the fleet costs on both networks. Furthermore, we assess the effect of changing cost parameters and satellite locations in the proposed IWLT system–analyzing system behavior and suggesting potential improvements–and evaluate several system alternatives in city logistics–consisting of different transportation network designs (single- and two-echelon), vehicle types, and operational constraints. On average, the proposed two-echelon IWLT system reduces the number of kilometers traveled by vehicles at street level by ranging from 20% to 30% compared to a typical single-echelon service design that relies solely on trucks.

A Multi‐Trip Vehicle Routing Problem With Release Dates and Interrelated Periods

ABSTRACTThis article proposes a new multi‐trip vehicle routing problem variant, originally motivated by a practical application, namely a car components distribution to repair centers (warehouses). The problem is named the multi‐trip vehicle routing with release dates and interrelated periods (MTVRP‐RDIP). The routes may start at different pre‐defined periods, called departure periods, and may have different durations. Thus, two routes that start at different periods may be active at the same period, leading to the so‐called interrelated periods. Moreover, a route may only start at a period if a vehicle is available, and the availability of the vehicle depends not only on the existing vehicles but also on the active routes at that time period. The clients are classified according to their importance and represent warehouses that require car components that are made available throughout the time horizon, thus leading to the consideration of release dates. Delays in satisfying client orders and not serving orders in the time horizon result in penalty costs that must be minimized. The objective to minimize also includes routing costs and vehicle utilization costs. Two metrics are defined to compute an order's delay, each leading to a different mixed integer linear model. Computational results showed that one model clearly outperforms the other and even this one is only suited to address the smallest instances. A matheuristic based on a rolling‐horizon process that iteratively solves the better‐performing model with fewer periods was designed to tackle the largest instances. The matheuristic can provide feasible solutions for all test instances in an efficient manner that are, on average, better than the ones provided by the model in most of the compared key performance indicators.

Data-driven collaborative healthcare resource allocation in pandemics

Severe shortages of healthcare resources are major challenges in pandemics, especially in their early stages. To improve emergency management efficiency, this paper proposes a novel rolling predict-then-optimize framework that includes three interactive modules, i.e., data-driven demand prediction, healthcare resource allocation, and parameter rolling update. Such a framework uses historical data to dynamically update the control parameters of the proposed Net-SEIHRD model, which predicts the healthcare needs of each region by jointly considering government interventions and cross-regional travel behaviors. Based on the forecasted healthcare resource demand in real-time, an optimization model is then formulated to realize coordinated resource allocation across multiple regions by minimizing the total generalized cost. To facilitate model solving, the proposed mixed integer nonlinear programming model is converted into an equivalent mixed integer linear model by using some linearization techniques. Finally, the proposed method is applied to the SARS-CoV-2 emergency response and collaborative allocation of healthcare resources in Shanghai, China. The results show that the proposed prediction model can effectively predict the peak and scale of the spread of the virus. Compared with the traditional LM and SEIHR models, the prediction accuracy of the Net-SEIHRD model is improved by 10.76% and 24.11%, respectively. Moreover, coordinated relief activities across regions, such as patient transfer and drug-sharing can improve the efficiency of pandemic control and save social costs.

Optimal Placement and Sizing of Distributed Generation in Electrical DC Distribution Networks Using a Stochastic Mixed-Integer LP Model

AbstractThis paper presents a stochastic mixed-integer linear mathematical model for finding the optimal placement and sizing of distributed generation in a DC distribution network, considering the uncertainty of electrical demand and distributed renewable sources. The proposed model accurately represents the original mixed-integer nonlinear model, obtaining a globally optimal solution in less computational time with low errors. The mathematical model allows for considering constraints related to the maximum limits for the penetration of distributed generation, such as those specified by Resolution CREG 174 of 2021. Furthermore, the uncertainties of the electrical demand, wind energy-based distributed generation (DG), and solar energy-based DG are considered in the mathematical models using a two-stage stochastic programming approach. The accuracy and efficiency of the proposed model were tested and validated on a 21-node DC test system from the specialized literature, and the effectiveness and robustness were assessed on a 69-node DC test system. The obtained results show that the proposed stochastic mixed-integer linear mathematical model performs well.

A comparison of 4th and 5th generation thermal networks with energy hub

State-of-the-art thermal networks are key to address decarbonization of the heating and cooling in buildings. Energy Hub considers synergy between elements and allows rapid comparison between district energy systems incorporating 4th generation (4G) and 5th generation (5G) thermal networks at early-stage designs. To better understand and quantify the differences between systems based on 4G and 5G, a mixed-integer linear model distinguishing specific features of both technologies is implemented. We find systems relying on 5G to perform environmentally and financially better than with 4G generation over a wide set of scenarios. This is due to the warm/cold coupling characterizing 5G technologies. Systems based on 5G can also reduce its carbon emissions more than those with 4G. However, performances with both technologies appear sensitive to the topology and location of central energy station.

A scenario-based decision framework for the order promising process in hybrid MTS/MTO production systems considering product substitution

ABSTRACT Considering the increasing competition among manufacturers and the diversification of customer orders, it is important to use the optimal production system. Hybrid manufacturing systems have proven to be a viable solution to achieve a satisfactory level of customization while maintaining adequate production volumes. This article determines the optimal strategy for decision makers in order to choose the appropriate method of supplying customers’ orders. In this article, a hybrid MTS-MTO production system has been studied in two different scenarios. For this purpose, a mixed-integer linear bi-objective model with product substitution and replaceable materials has been developed and solved by a Lagrangian Relaxation algorithm. The findings of this study reveal that enabling product substitution by manufacturers leads to an increase in the number of accepted orders, along with resulting in a higher average level of customer dissatisfaction. The system prioritizes the fulfillment of orders from permissive customers. Also, with the increase of uncertainty in each of the supplying methods, manufacturers are using other sources to supply raw materials to satisfy the accepted orders.

The Dial-a-Tour Problem

This article delves into the Dial-a-Tour Problem (DATP) frequently encountered in local travel agencies, known for their personalized and hands-on customer service. In the DATP, multiple tourist groups are efficiently matched with different vehicles using vanpooling, maximizing vehicle occupancy while ensuring seamless transfers to the local attractions included in their booked tours. Each tourist group is picked up from and returned to their respective hotels after their journey. The predetermined locations and the sequence of visits for each tour are known in advance. Therefore, the primary objective is to minimize the number of vehicles required to fulfill all requests and maintain the desired level of service.To tackle this challenge, the article presents a comprehensive approach. It introduces a mixed-integer linear model, two constraint programming formulations, and a novel operator called the k-buffer insertion operator. This operator serves as the centerpiece of a descent heuristic proposed in the study.Extensive evaluation across a diverse range of randomly generated instances underscores the superiority of the proposed constraint programming formulations over the mixed integer linear programming model. However, it is noteworthy that the proposed heuristic surpasses even the most effective constraint programming formulation. This highlights its remarkable efficiency in solving real-size instances within notably brief computation times. Such prowess underscores the practical utility of this heuristic in effectively tackling the complexities inherent in the Dial-a-Tour Problem.

Data-driven robust optimisation of hydrogen infrastructure planning under demand uncertainty using a hybrid decomposition method

In the race towards “Net-zero”, hydrogen has emerged as one of the key alternatives to carbon-based fossil fuels for a sustainable decarbonisation. This work studies the spatially explicit multi-period hydrogen infrastructure planning under demand uncertainty that contributes to the heat decarbonisation in Great Britain. Demand uncertainty surrounding future hydrogen supply chains poses challenges to cost optimisation and system security, so uncertainty-resilient policies are required to ensure robust operations. In this work, we employ data-driven robust optimisation to develop a framework for uncertainty-aware representative days explicitly characterised by polyhedral uncertainty sets. The proposed framework is applied on a multi-period mixed-integer linear model with dual temporal resolution which aims to determine the optimal yearly investment decisions and hourly operational decisions for the hydrogen infrastructure planning under demand uncertainty. To efficiently solve the large-scale two-stage adaptive robust optimisation problem, a hybrid decomposition algorithm is developed based on a two-step hierarchical procedure and the column-and-constraint generation method, which can significantly reduce the computational complexity. The optimisation results highlight how uncertainty can result in the total cost increase, and verify the advantages on controlling solution conservatism in the adaptive robust optimisation compared to the static robust optimisation.

Design of closed-loop manufacturing-remanufacturing system network considering uncertain defect and waste rate: Scenario-based chance-constrained programming

To facilitate sustainability by extending the life-cycle and residual value of production materials, a collaborate scheduling scheme for closed-loop manufacturing system with disturbance uncertainty is studied. Based on the features of the closed-loop manufacturing system under multi-product and multi-period scenarios, a mixed-integer linear optimization model is developed, where manufacturing and remanufacturing units are coordinated by recycling the production materials from work-in-process and final products. Based on this, a novel closed-loop manufacturing-remanufacturing scheduling framework is proposed. Additionally, in response to variations in product quality and processing capabilities, a scenario-based chance-constrained programming is introduced to address potential sources of uncertainty. Subsequently, a sequential optimization method is devised to reduce computational complexity. Finally, an industrial case from the electronic factory is investigated to demonstrate its applicability by implementing the proposed approach with comparison to several other strategies.

Optimizing the organization of the first mile in agri-food supply chains with a heterogeneous fleet using a mixed-integer linear model

Consumers are increasingly demanding high-quality food, which presents significant challenges for agricultural supply chains. While the majority of research in the agri-food sector has concentrated on optimizing logistics costs and meeting demand by focusing on minimizing the last mile, the complexity of the first mile in the agricultural supply chain has been less explored. Farmers must efficiently manage the harvesting process and the transportation of harvested produce to consolidation centers to ensure the delivery of high-quality products. This paper addresses this research gap by introducing a mixed-integer programming model that leverages vehicle routing problem concepts to optimize the logistics processes involved in transporting harvested products from various fields to a central depot. The primary objective is to minimize total logistics costs associated with visiting different fields during a pick-up round using multiple vehicles. The model has been applied to a case study involving an agricultural cooperative in Greece as part of the European BBTWINS project, which aims to enhance agri-food value chain digitalization for improved resource efficiency. The results demonstrate that organizing the first mile of the agri-food supply chain with a cooled vehicle for pick-up rounds can reduce logistics costs by up to 40% compared to the current practices.

Low-carbon operation strategy of virtual power plant considering progressive demand response

Adapting virtual power plants (VPPs) to the development of new power systems has gradually become a trend. Meanwhile, the economical and effective use of demand-side resources in VPPs has become a research hotspot in the field of new energy. This paper proposes a low-carbon operation strategy with progressive demand response (DR). A DR model id established by evaluating the scheduling potential of low-carbon and flexible loads and the DR transfer of different users motivated by incentive mechanisms. Because the changing carbon price in carbon trading also affects the response transfer amount, it is beneficial to reduce the operating cost of the energy system by using consumers’ surplus value according to consumer psychology to guide energy users to participate in DR. Finally, wind, light, user load data and VPP internal unit output are optimized in rolling mode over multiple time scales, and the optimal adjustment results of the previous day are taken as the basis. At the same time, the output deviation penalty function is added to further reduce the system cost. The nonlinear model of DR cost is transformed into a mixed integer linear model, which is modeled on MATLAB software and solved by a CPLEX optimization solver. The case analysis shows that the progressive DR strategy may increase the total operating cost of the system, but reduce the cost of the carbon emission. The proposed model and method can not only improve the feasibility and effectiveness of market supervision strategy, but also provides a useful reference for operators and demand-side users in energy systems to choose strategies.

A dynamic bi-objective optimization model for a closed loop supply chain design under environmental policies

Given the increasing focus on sustainability and environmental policy constraints, companies are required to redesign their supply chains. This paper explores the optimization of a closed loop supply chain (CLSC) network under both economic and environmental considerations. To achieve this, a bi-objective mixed integer linear model was developed. The proposed model identifies the optimal selection of CLSC facilities and manages both forward and reverse flows between them. The economic objective is reached by minimizing the total CLSC costs, while the environmental objective is satisfied by reducing CO2 emissions throughout the network. Products can be returned throughout their entire life cycle, which is why our model incorporates a dynamic aspect by considering product life cycle phases as time periods for the decision horizon. The model was tested through numerical experiments using a meta-heuristic approach based on the non-dominated sorting genetic algorithm NSGA-II. This algorithm produces a set of Pareto-optimal solutions that balance both objectives effectively. The results showed good performance in terms of computational time and optimization. Pareto solutions offered various options for managers and decision makers aiming for a sustainable closed loop supply chain design.

A multi-objective fuzzy mathematical model for circular economy with leasing as a strategy

PurposeThis paper aims at developing a multi-objective mathematical model of circular economy that integrates key concept of leasing as a strategy in addition to reuse, refurbishing, primary recycling, secondary recycling and disposal.Design/methodology/approachThis paper proposes multi-objective fuzzy mixed integer linear programming mathematical model considering multi-product, multi-echelon and multi-capacitated concepts of the circular economy. The three objectives of the proposed model, namely, economic, environmental and social are solved simultaneously using constraint approach to obtain balanced trade-off between the objective functions. The model is validated by solving a case study from the literature. The proposed model is made pragmatic for industrial application by considering multi-external suppliers multi-customer zones, multi-disassembly centers, multi-collection centers and multi-refurbishing centers and accounting for purchasing, processing, transportation, set-up costs and capacity constraints at the same time.FindingsThe results show that the leasing of the products improves the economic function in addition to the known environmental improvements. The proposed model also shows that the circular economy can generate the jobs for the unskilled people at different locations.Research limitations/implicationsThe proposed model can be further improved by considering the non-linearity due to economy of scale at various centers and in transportation. The model can be further extended to make it multi-period model.Practical implicationsThe proposed model of circular economy can be used by the organizations as a policy tool to decide the optimum number of collection centers, disassembly centers, refurbishing centers, recycling centers and disposal centers and their optimum locations and allocations. The organizations can also trade-off among economic, environmental and social benefits of their proposed decisions in circular economy.Originality/valueThe originality of the proposed mathematical model is consideration of leasing as a strategy to have better control over the supply chain for circularity; considering the training of unskilled people for backward supply chain jobs and accounting for primary recycling and secondary recycling separately for economical computation.

Mathematical Programming Formulations for the Berth Allocation Problems in Container Seaport Terminals

Background: Improving the performance of marine terminals is one of the major concerns of both researchers and decision-makers in the maritime transportation sector. The problem of container storage planning and the berth allocation problem (BAP) are the two mainstays of optimizing port operations. Methods: In this work, we address these two issues, proposing two mathematical models that operate sequentially and are applicable to both static and dynamic cases. The first developed model is a mixed-integer linear problem model aimed at minimizing vessel traffic time in the port. The second model developed is a multi-objective optimization model based on goal programming (GP) to minimize both container transfer time and the number of storage areas (minimizing container dispersion). Results: The robustness of the proposed models has been proven through a benchmark with tests using data from the literature and real port data, based on the IBM ILOG CPLEX 12.5 solver. Conclusions: The two developed mathematical models allowed the both minimization of the transfer time and the number of used storage areas, whatever the number of operations handling companies (OHCs) operating in the seaport and for both static and dynamic cases. We propose, as prospects for this work, the development of a heuristic model to deal with the major instances relating to the case of large ports.

Multi-objective optimization for perishable product dispatch in a FEFO system for a food bank single warehouse

One of the main challenges of food bank warehouses in developing countries is to determine how to allocate perishable products to beneficiary agencies with different expiry dates while ensuring food safety, meeting nutritional requirements, and minimizing the shortage. The contribution of this research is to introduce a new multi-objective, multi-product, and multi-period perishable food allocation problem based on a single warehouse management system for a First Expired-First Out (FEFO) policy. Moreover, it incorporates the temporal aspect, guaranteeing the dispatch of only those perishable products that meet the prescribed minimum quality standards. A weighted sum approach converts the multi-objective problem of minimizing a vector of objective functions into a scalar problem by constructing a weighted sum of all the objectives. The problem can then be solved using a standard constrained optimization procedure. The proposed mixed integer linear model is solved by using the CPLEX solver. The solution obtained from the multi-objective problem allows us to identify days and products experiencing shortages. In such cases, when there is insufficient available inventory, the total quantity of product to be dispatched is redistributed among beneficiaries according to a pre-established prioritization. These redistributions are formulated as integer programming problems using a score-based criterion and solved by an exact method based on dynamic programming. Computational results demonstrate the applicability of the novel model for perishable items to a real-world study case.

A machine learning approach to analysis of Canadian provincial power system decarbonization

Canada committed to reducing national greenhouse gas emissions by 40–45 % by 2030 and achieving net-zero by 2050, relying on low-carbon electrification across economic sectors. This study investigates the machine learning’s potential to analyze least-cost optimization of Canadian provincial power systems to 2050, to identify robust pathways for net-zero electricity system greenhouse gas emissions, and to provide policy insights respecting wind and solar generation, reducing emissions from natural gas generation, and expanding interprovincial transmission. Surrogate modeling is applied as sensitivities to generation technology capital costs, electricity demand, and carbon pricing to results from an existing capacity expansion mixed-integer linear optimization model. Under almost all future scenarios, six provinces develop new wind generation, with substantial deployment in Alberta (mean 27.3 GW, max. 62.7 GW), Ontario (mean 6.8 GW, max. 19.3 GW), Saskatchewan (mean 3.6 GW, max 8.9 GW) and Manitoba (mean 1.5 GW, max 3.5 GW). With the potential exception of BC, where higher-cost wind generation results in solar generation becoming the primary low-carbon electrical energy source (mean 25.2 GW), future solar generation across Canada is more limited (mean 37.1 GW) and less certain than wind generation in all other Canadian provinces. Existing natural gas generation is maintained in almost all scenarios in all provinces, providing support for maintaining these resources for reliability and reserve requirements. Very limited natural gas is developed under any scenarios, with the exception of British Columbia (mean 4.3 GW) where more than 80 % of scenarios include new natural gas capacity. Additional transmission transfer capacity is developed in most scenarios between BC-Alberta (∼60 % of scenarios), Saskatchewan-Manitoba (∼85 %), and Ontario-Quebec (∼75 %). Carbon prices, along with wind and solar capital costs, act as primary determinants of least-cost decarbonization pathways. These findings offer insights for Canadian policymakers into net-zero pathways for provincial electricity systems that remain robust despite variability in inputs.

Optimizing the Selection of Mass Vaccination Sites: Access and Equity Consideration.

In the early phases of the COVID-19 pandemic, vaccine accessibility was limited, impacting large metropolitan areas such as Los Angeles County, which has over 10 million residents but only nine initial vaccination sites, which resulted in people experiencing long travel times to get vaccinated. We developed a mixed-integer linear model to optimize site selection, considering equitable access for vulnerable populations. Analyzing 277 zip codes between December 2020 and May 2021, our model incorporated factors such as car ownership, ethnic group disease vulnerability, and the Healthy Places Index, alongside travel times by car and public transit. Our optimized model significantly outperformed actual site allocations for all ethnic groups. We observed that White populations faced longer travel times, likely due to their residences being in more remote, less densely populated areas. Conversely, areas with higher Latino and Black populations, often closer to the city center, benefited from shorter travel times in our model. However, those without cars experienced greater disadvantages. While having many vaccination sites might improve access for those dependent on public transit, that advantage is diminished if people must search among many sites to find a location with available vaccines.

Integrated corridor management by cooperative traffic signal and ramp metering control

AbstractThis paper formulates a cooperative traffic control methodology that integrates traffic signal timing and ramp metering decisions into an optimization model to improve traffic operations in a corridor network. A mixed integer linear model is formulated and is solved in real time within a model predictive controller framework, where the cell transmission model is used as the system state predictor. The methodology is benchmarked in a case study corridor in San Mateo, CA, in VISSIM with two optimization scenarios, namely, optimal metering and optimal signal control, and two simulation scenarios with a preset metering plan and no metering. The numerical results show that integrated traffic signal and ramp metering control reduces delays, stops, and travel times of the corridor by up to 33.1%, 36%, and 16.4%, respectively, compared to existing benchmark conditions. With appropriate weights prioritizing freeway or arterial street operations, the integrated control balances traffic congestion between the arterial street and the freeway.

Optimal sizing of multi-energy microgrid with electric vehicle integration: Considering carbon emission and resilience load

To address the intermittency of renewable energy sources, global warming, and increasing load demands, this paper proposes the optimal sizing of a multi-energy microgrid (MEMG) consisting of electrical, thermal, cooling, and hydrogen networks. The system integrates multi-energy storage and EVs along with the resilience load to facilitate a robust operation scheme. The paper introduces an improved resilience backup mechanism for EVs using hybrid storage. Then, random samples for stochastic parameters are generated with Monte Carlo Simulations (MCS). To this end, a mixed-integer linear programming-based model is developed to minimize cost, emissions, and load shed. Then, a CPLEX solver is applied to solve it efficiently. The optimal MEMG with the hydrogen network reduces cost by 4% and emissions by 40%. The case study validates that hybrid storage (BES-HST-TST) can effectively reduce the electricity grid purchase to zero, making MEMG self-sufficient while yielding the least annual system cost (ASC) of 3855562$ and decreasing emissions to 8385 kg, resulting in economic savings, environmental sustainability, and increased utilization of renewables. Notably, V2G can save 0.7546% of MEMG extra cost incurred due to EVs integration and significantly reduces CO2 emissions by 4%. A novel finding is that the proposed hybrid storage backup mechanism can effectively minimize the extra cost of keeping the resilience load by retaining 31.46% of backup in hydrogen form. It mitigates risk associated with power outages while achieving the least ASC of 3858933$ and CO2 emissions of 8043 kg. These results can help realize a green, cost-effective, and efficient energy system.