Mixed Integer Linear Programming Model Research Articles

Optimizing Supply Chain Design under Demand Uncertainty with Quantity Discount Policy

In typical business situations, sellers usually offer discount schemes to buyers to increase overall profitability. This study aims to design a supply chain network under uncertainty of demand by integrating an all-unit quantity discount policy. The objective is to maximize the profit of the entire supply chain. The proposed model is formulated as a mixed integer nonlinear programming model, which is subsequently linearized into a mixed integer linear programming model and hence able to obtain a global solution. Numerical examples in the manufacturing supply chain where customer demand follows normal distributions are used to assess the effect of quantity discount policies. Key findings demonstrate that the integration of quantity discount policies significantly reduces total supply chain costs and improves inventory management under demand uncertainty, and decision makers need to decide a balance level between service levels and profits.

Augmented ɛ‐constraint‐based matheuristic methodology for Bi‐objective production scheduling problems

AbstractMatheuristic is an optimisation methodology that integrates mathematical approaches and heuristics to address intractable combinatorial optimisation problems, where a common framework is to insert mixed integer linear programming (MILP) models as local search functions for evolutionary algorithms. However, since a mathematical programming formulation only tries to find the solution with the best objective value, matheuristics are rarely adopted to multi‐objective scenarios asking for a set of Pareto optimal solutions, for example, vehicle routing problems and production scheduling problems. In this situation, the ɛ‐constraint, which transforms multi‐objective problems into single‐objective formulations by considering selected objectives as constraints, seems to be a promising approach. First, an augmented ɛ‐constraint‐based matheuristic methodology (ɛ‐MH) is proposed to apply the idea of ɛ‐constraint to embedded MILP models, so that Pareto fronts obtained by meta‐heuristics can be further improved by solving a set of MILP models. Afterwards, four speed‐up strategies are developed to alleviate the computational burden resulting from repeatedly solving mathematical formulations, which also imply preferable scenarios for taking advantages of the ɛ‐MH. Finally, several real‐world bi‐objective scheduling problems are discussed to present potential applications for the proposed methodology.

Deep reinforcement learning driven trajectory-based meta-heuristic for distributed heterogeneous flexible job shop scheduling problem

As the production environment evolves, distributed manufacturing exhibits heterogeneous characteristics, including diverse machines, workers, and production processes. This paper examines a distributed heterogeneous flexible job shop scheduling problem (DHFJSP) with varying processing times. A mixed integer linear programming (MILP) model of the DHFJSP is formulated with the objective of minimizing the makespan. To solve the DHFJSP, we propose a deep Q network-aided automatic design of a variable neighborhood search algorithm (DQN-VNS). By analyzing schedules, sixty-one types of scheduling features are extracted. These features, along with six shaking strategies, are used as states and actions. A DHFJSP environment simulator is developed to train the deep Q network. The well-trained DQN then generates the shaking procedure for VNS. Additionally, a greedy initialization method is proposed to enhance the quality of the initial solution. Seven efficient critical path-based neighborhood structures with three-vector encoding scheme are introduced to improve local search. Numerical experiments on various scales of instances validate the effectiveness of the MILP model and the DQN-VNS algorithm. The results show that the DQN-VNS algorithm achieves an average relative percentage deviation (ARPD) of 3.2%, which represents an approximately 88.45% reduction compared to the best-performing algorithm among the six compared, with an ARPD of 27.7%. This significant reduction in ARPD highlights the superior stability and performance of the proposed DQN-VNS algorithm.

Stochastic Risk-driven Bidding of a Solar and Storage Aggregator in Primary Frequency and Energy Markets: A Performance-based Capacity Allocation Approach

The ever-increasing impact of deploying renewable energy resources reducing power system inertia, requires distributed energy resource (DER) aggregators to secure high-performance, fast-responding primary frequency reserve (PFR) in ancillary service markets. However, enabling immediate and local regulation of the operating point upon detecting frequency deviations imposes the necessity of allocating specific headroom for DERs. The individual headroom susceptibility to uncertainty and response speed of a certain DER not only compromises aggregated headroom performance in the PFR market but also leads to incurring unnecessary opportunity costs in the energy market and inadequate frequency control in the power system. To address this challenge, this paper introduces an integrated probabilistic performance index for DERs that considers their individual interactive heterogeneous response speed and uncertainty. Using this index, aggregators overseeing photovoltaic and smart buildings equipped with energy storage can evaluate, allocate, and remunerate optimal headroom capacity for individual DERs based on their performance. This approach ensures maximum aggregated profits and minimum opportunity costs in the PFR and energy markets while improving the performance of the provided PFR capacity, respectively. These tasks are aimed by a novel optimal capacity allocation strategy for an aggregator in both markets. This strategy employs a mixed-integer linear programming model to find the optimal solution and the conditional value-at-risk measure to tackle the uncertainties faced with the problem. The efficiency of the proposed model is assessed within a system reflecting the ERCOT market structure, demonstrating improved aggregated headroom performance, reduced energy opportunity costs, diminished risk of capacity shortage, and improved load frequency control (LFC) from the power system perspective.

Sustainable optimization of balancing valve settings in urban heating systems with an enhanced Jaya algorithm

With the continuous growth of global energy demand, energy conservation in heating has become a research hotspot for urban sustainable development. However, the research on energy conservation by adjusting the balance valve in the secondary network of heating system is relatively few. To address this gap, this paper solves the balancing valve opening adjustment (BVOA) problem with the goal of minimizing heat consumption and the energy usage of circulation pumps. A mixed-integer linear programming model is designed for this problem, and an effective EJaya algorithm is proposed. Initially, a random initialization method with three scrambling strategies is employed to generate high-quality solutions. Moreover, a segmented operator is proposed to enhance the algorithm’s search efficiency. Subsequently, Q-learning is utilized to select from three neighborhood operators, enhancing the algorithm’s local search capabilities. Additionally, an adaptive search balance strategy is designed to balance the methods of updating solutions, aiding in finding better solutions. Finally, extensive experiments have validated the effectiveness and efficiency of the proposed algorithm in solving the BVOA problem, contributing to the promotion of urban energy sustainability and social welfare.

A sustainable network design for municipal solid waste management considering waste-to-energy conversion under uncertainty

This research proposes a sustainable municipal solid waste management (MSWM) system to address population growth, consumerism, and resource scarcity. It introduces a two-phase decision structure combining multi-attribute decision-making (MADM) tools. Phase one develops a hybrid risk-oriented fuzzy MADM tool for selecting optimal waste-to-energy technologies, considering environmental and technological factors. Phase two uses a fuzzy bi-objective multi-period mixed-integer linear programming model to design the MSWM supply chain efficiently under uncertainty. A case study in North Tehran demonstrates the framework’s practical applicability and effectiveness in real-world scenarios, highlighting that effective source separation boosts recycling, reduces costs, and provides social benefits. Sensitivity analyses offer insights to enhance waste segregation practices. The study emphasizes integrating economic, environmental, and social sustainability in waste management decision-making. By offering a novel, holistic approach, this research addresses existing gaps in systematic decision-making processes and provides a robust tool for municipalities to optimize strategies under uncertainty.

Efficient Algorithm for Generating Optimal Inequality Candidates for MILP Modeling of Boolean Functions

Mixed-Integer Linear Programming (MILP) modeling has become an important tool for both the analysis and the design of symmetric cryptographic primitives. The bit-wise modeling of their nonlinear components, especially the S-boxes, is of particular interest since it allows more informative analysis compared to word-oriented models focusing on counting active S-boxes. At the same time, the size of these models, especially in terms of the number of required inequalities, tends to significantly influence and ultimately limit the applicability of this method to real-world ciphers, especially for larger number of rounds. It is therefore of great cryptographic significance to study optimal linear inequality descriptions for Boolean functions. The pioneering works of Abdelkhalek et al. (FSE 2017), Boura and Coggia (FSE 2020) and Li and Sun (FSE 2023) provided various heuristic techniques for this computationally hard problem, decomposing it into two algorithmic steps, coined Problem 1 and Problem 2, with the latter being identical to the well-known NP-hard set cover problem, for which there are many heuristic and exact algorithms in the literature. In this paper, we introduce a novel and efficient branch-and-bound algorithm for generating all minimal, non-redundant candidate inequalities that satisfy a given Boolean function, therefore solving Problem 1 in an optimal manner without relying on heuristics. We furthermore prove that our algorithm correctly computes optimal solutions. Using a number of dedicated optimizations, it provides significantly improved runtimes compared to previous approaches and allows the optimal modeling of the difference distribution tables (DDT) and linear approximation tables (LAT) of many practically used S-boxes. The source code for our algorithm is publicly available as a tool for researchers and practitioners in symmetric cryptography.

Optimizing customized bus services for multi‐trip urban passengers: A bi‐objective approach

AbstractCustomized bus services typically focus on single‐trip requests, which often struggle to accommodate the growing needs for varied multiple trips in urban daily travel. This paper addresses the customized bus routing problem for passengers with multiple trips. A bi‐objective mathematical model is established for maximizing the operational profit and minimizing the travel costs by considering the characteristics of the multi‐trip requests and time‐dependent travel time. Besides, a novel profit objective function is proposed considering the service's completion status and the starting price. Since the proposed mixed integer linear programming model is an NP‐hard problem, a non‐dominated sorting genetic algorithm II‐based method is proposed to handle different sizes of instances. Finally, the instances with multi‐trip requests are carried out to test the accuracy of the model and the effectiveness of our method compared with Gurobi and the local search‐based multi‐objective algorithm approach.

Electrification of transportation: A hybrid Benders/SDDP algorithm for optimal charging station trading

This paper examines the electrification of transportation as a response to environmental challenges caused by fossil fuels, exploring the potential of battery electric vehicles and hydrogen fuel cell vehicles as alternative solutions. However, a significant barrier to their widespread adoption is the limited availability of charging infrastructure. Therefore, this study proposes the development of comprehensive charging stations capable of accommodating both battery and hydrogen vehicles to address this challenge. The energy is purchased from the day-ahead and intraday auction-based electricity markets, where the electricity price is subject to uncertainty. Therefore, a two-stage stochastic programming model is formulated while the price scenarios are generated utilizing a k-means clustering algorithm. Given the complexity of the proposed model, an efficient solution approach is developed through the hybridization of the Benders decomposition algorithm and stochastic dual dynamic programming. In the Benders master problem, day-ahead bidding variables are determined, whereas the Benders sub-problem addresses intraday bidding and charging station scheduling variables, employing stochastic dual dynamic programming to tackle its intractability. Additionally, we transform the mixed integer linear program model of the second stage problem into a linear program, confirming its validity through KKT conditions. Our model provides practical insights for making informed decisions in electricity markets based on sequential auctions. While the bidding curves submitted to the day-ahead market remain unaffected by scenarios, those submitted to the intra-day market show dependence on fluctuations in day-ahead market prices.

A reinforcement learning-based multiobjective heuristic algorithm for multiple-truck routing problems with heterogeneous drones

As a result of new technological developments, drone usage has become common in last-mile logistics activities. Owing to the limited flight range of drones, drones may service customers by using trucks as hubs. Although the truck routing problem with drones (TRP-D) has been studied frequently, studies that have addressed this problem in a multiobjective manner are rare. In this study, multiobjective TRP-D is discussed. In this problem, multiple trucks are considered. Each truck is equipped with multiple heterogeneous drones. A multiobjective mixed-integer linear programming (MILP) model is proposed. Since the problem is NP-hard, a reinforcement learning (RL)-based multiobjective heuristic algorithm is proposed to obtain solutions in large-scale cases. A three-phase heuristic feasible-solution-generation algorithm is proposed. The variable neighbourhood descent (VND) algorithm with RL is proposed as a local search algorithm for the multiobjective optimization problem. The results of the proposed heuristic algorithm are compared with the Pareto optimal solutions for small-sized test problems via the augmented ε-constraint (AUGMECON) method and the CPLEX solver. The proposed algorithm is also compared with state-of-the-art multiobjective heuristic algorithms for large-scale problems. According to the results, the proposed algorithm is an efficient multiobjective heuristic algorithm for the addressed problem.

Applying the Simulated Annealing Algorithm to the Set Orienteering Problem with Mandatory Visits

This study addresses the set orienteering problem with mandatory visits (SOPMV), a variant of the team orienteering problem (SOP). In SOPMV, certain critical sets must be visited. The study began by formulating the mathematical model for SOPMV. To tackle the challenge of obtaining a feasible route within time constraints using the original MILP approach, a two-stage mixed-integer linear programming (MILP) model is proposed. Subsequently, a simulated annealing (SA) algorithm and a dynamic programming method were employed to identify the optimal route. The proposed SA algorithm was used to solve the SOP and was compared to other algorithms, demonstrating its effectiveness. The SA was then applied to solve the SOPMV problem. The results indicate that the solutions obtained using SA are superior and more efficient compared to those derived from the original MILP and the two-stage MILP. Additionally, the results reveal that the solution quality deteriorates as the ratio of the set of mandatory visits increases or the maximum allowable travel time decreases. This study represents the first attempt to integrate mandatory visits into SOP, thereby establishing a new research direction in this area. The potential impact of this research is significant, as it introduces new possibilities for addressing complex combinatorial optimization problems.

Optimal cost predictive BMS considering greywater recycling, responsive HVAC, and energy storage

A crucial aspect of a sustainable city is ensuring that energy and water supplies can adequately meet urban demand. With the scarcity of natural resources and growing electricity and water demand, it becomes increasingly important for consumers to manage their resource usage more efficiently. This paper proposes a novel perspective of demand-side management coordination strategy for a building's water-energy nexus to enhance the resilience and efficiency of the overall electricity-water-heating system. The model is formulated to optimally coordinates the onsite greywater recycling system, heating, ventilation, air conditioning (HVAC) loads, and distributed energy generation systems with a bidirectional grid connection in a residential building. All subsystems are controlled by a model predictive controller (MPC) receiving real-time time of use (ToU) pricing from electricity and water utilities. The presented mixed integer linear programming model is verified to meet the customers' demands while reducing the operational costs. Results are compared with benchmark systems lacking the water recycling or energy storage system showing 8.3 % operational cost reduction while reducing potable water consumption by 21.5 %. The effect of increased MPC control horizon is also studied showing reduction in cost with increased horizon. Detailed analysis of the proposed framework computational burden and effect of prediction errors is performed to prove the MPC adaptability and robustness. Testing under increased room size and different user preferences further validate the efficacy of the proposed scheme in reducing the operational costs.

Dynamic airport gate assignment with improved Shuffled Frog-Leaping Algorithm and triangle membership function

he rapid development of the air transportation industry has increased air traffic, posing challenges to the task of airport gate assignment (AGA) for flights. Most past studies have solved the AGA problem (AGAP) using deterministic models, which are incapable of dealing with uncertainty and dynamic conditions at airports. Thus, this research employs fuzzy theory and proposes a triangular membership function to handle flight uncertainty in the AGAP. In addition, an improved metaheuristic, termed the improved Shuffled Frog-Leaping Algorithm (ISFLA), is proposed to circumvent the computationally intractable problems commonly faced by exact approaches when handling large instances. In this research, the AGAP is first formulated as a stochastic Mixed-Integer Linear Programming (MILP) model, with stochastic flight lateness and earliness considered. The objective of this model is to minimize the total cost, which consists of three sub-costs: passenger walking distances, non-preferred gate (NPG) assignments for planes, and fuzzy idle times of gates. These three sub-costs correspond to the major concerns of passengers, airlines, and airports, respectively. The cooperation between the ISFLA and the triangular membership function demonstrates their capability to effectively handle big AGAP instances. Furthermore, the experimental results show that the ISFLA outperforms the standard SFLA, Genetic Algorithm (GA), Particle Swarm Optimization (PSO), and Firefly Algorithm (FA).

Customer order scheduling on a serial-batch machine in precast bridge construction

Prefabricated construction is widely adopted in the current bridge construction project, especially offshore bridges. The realistic requirements of large quantities of prefabricated parts and tight delivery schedules make it extremely challenging to develop optimal scheduling. We address a new customer order scheduling on a serial-batch machine (COS-SBM) to reduce the sum of inventory holding costs of finished jobs and tardiness costs of orders in precast bridge construction. In the COS-SBM problem, all jobs with incompatibility in orders need to be divided into batches, which are then scheduled for processing on a serial-batch machine. We develop a mixed-integer linear programming model to formulate this new problem. Since the COS-SBM problem is NP-hard, we propose a genetic algorithm based on a novel batch sequencing and forming encoding method (GA-BSFE), which makes the scheduling and batching decisions simultaneously to enhance its exploration. Moreover, we design an efficient three-stage heuristic based on the order weighted modified due date rule and batch weighted longest processing time rule. The three-stage heuristic is introduced into the initiation of GA-BSFE to enhance its exploitation. Finally, a set of instances generated based on the realistic production of precast girders is tested to validate the effectiveness of GA-BSFE. The performance analysis suggests that GA-BSFE is the most appropriate for the COS-SBM problem.

Collaboration and resource sharing in the multidepot time-dependent vehicle routing problem with time windows

Concerns about energy conservation and emission reduction have highlighted the importance of environmentally sound logistics networks in urban areas. These networks are deeply intertwined with urban traffic systems, where variations in transit speeds can significantly increase the energy consumption and carbon emissions of delivery vehicles, compromising the environmental sustainability of urban deliveries. To address this, we propose a multidepot time-dependent vehicle routing problem with time windows, enhancing route planning flexibility and resource configuration. Our approach begins with a route spatiotemporal decomposition method to estimate vehicle travel times and emissions based on varying vehicle speeds. We then develop a multiobjective mixed integer linear programming model that aims to minimize total operating costs, the number of vehicles, and carbon dioxide emissions. A hybrid heuristic algorithm combining spectral clustering, multiobjective ant colony optimization, and variable neighborhood search is proposed to solve the model. This algorithm incorporates collaboration and resource sharing strategies, a pheromone initialization mechanism, a novel heuristic operator that accounts for time dependency, and a self-adaptive update mechanism, enhancing both solution quality and algorithm convergence. We compare the performance of our algorithm with that of the CPLEX solver, multiobjective ant colony optimization, non-dominated sorting genetic algorithm-Ⅲ, and multiobjective particle swarm optimization. The results demonstrate the superior convergence, uniformity, and spread of our proposed algorithm. Furthermore, we apply our model and algorithm to a real-world case in Chongqing, China, analyzing optimized results for different time intervals and vehicle speeds. This study offers robust methodologies for theoretically and practically addressing the multidepot time-dependent vehicle routing problem with time windows, contributing to the development of economical, efficient, collaborative, and sustainable urban logistics networks.

Maximizing virtual power plant profit: A two-level optimization model for energy market participation

Managing dispersed generation via virtual power plants (VPPs) is crucial for maximizing profits in electricity markets. This paper presents a model aimed at maximizing VPP profit through participation in the energy market. The proposed model addresses grid and security constraints of units using deterministic programming, formulated as an equilibrium-constrained, two-level mathematical optimization model. The first level focuses on maximizing VPP profit, while the second optimizes social welfare. Applying duality theory transforms this two-level model into a mixed-integer linear programming model, further refined using Karush–Kuhn–Tucker (KKT) optimality conditions. Given the inherent conflict in these objectives, a novel algorithm employing water flow dynamics is proposed for solving the model. To enhance method performance, the Pareto criterion and fuzzy decision-making are incorporated. Model tests are conducted on a standard 24-bus IEEE grid, demonstrating its efficiency. For the single-objective problem without line congestion, the solving time was 12 s. Introducing line congestion increased the profit by 13.4 %, from $40,413.21 to $45,837.32. In the two-objective problem without congestion, the profit ranged between $36,928.72 and $42,813.28, and emissions ranged from 275.21 to 2,916.32 pounds. With congestion, the profit range increased by a maximum of 8.7 %, and emissions were reduced by up to 4.6 %.

Pickup and Delivery Vehicle Routing Problem with Hybrid Missions and Third-Party Vehicle Rentals: A Case Study of Animal Feed Industry

The vehicle routing problem for small and medium-sized animal feed manufacturers is a complex challenge because each transport route can vary significantly between delivering products to customers and picking up raw materials to return to the factory. Currently, businesses lack well-planned management for utilizing a limited number of trucks. As a result, they often incur additional costs by relying on third-party vehicle rentals. This research aims to reduce transportation costs for small and medium-sized animal feed manufacturers by solving a formulated mixed-integer linear programming (MILP) model, using experimental data that accurately reflects the actual problems faced. The MILP model results help reduce transportation costs by up to 18.5% compared to current practices by optimizing route sequences and incorporating the use of third-party vehicle rentals. This also serves as a guideline for reducing transportation costs in other similar industries.

Virtualized network functions resource allocation in network functions virtualization using mathematical programming

Network Functions Virtualization (NFV) revolutionizes network services by eliminating the need for dedicated hardware. This virtualization enables flexible and efficient deployment of various network functions like proxies, firewalls, and load balancers. Providing the service requested by the user in the network is done by a sequence of virtual network functions, which are known as service functions chain. One of the main challenges in the development of network functions virtualization architecture is the allocation of resources to the requested network services in network infrastructures, this challenge is called network function virtualization resource allocation problem. Therefore, this paper addresses the resource allocation problem in Network Functions Virtualization (NFV) architectures using mathematical programming techniques. A multi-objective mixed-integer linear programming (MILP) model is proposed to optimize resource allocation for virtual network functions (VNFs). The model incorporates constraints related to node and link resource capacities, as well as delay requirements. The objective functions focus on maximizing network throughput, minimizing node resource costs (CPU cores and memory), reducing capital and operational expenses, and ensuring efficient execution time. These constraints and objective functions are formally defined by mathematical functions. The proposed mathematical model is implemented and solved using the Cplex solver. To evaluate the effectiveness of the proposed mathematical model, various network topologies were evaluated under different parameters. These parameters included the length of Service Function Chains (SFCs), the number and length of flows, node resource capacities, the number of nodes and VNFs. The experimental results demonstrated the model’s ability to efficiently allocate resources to VNFs across these different scenarios.

Preemptive Software Project Scheduling Considering Personality Traits

The success of software projects is inseparable from the effective scheduling of multi-skilled human resources. In software development, employees’ personality traits often affect the performance of their work. Additionally, software developers are usually multi-tasking, which makes software development tasks preemptive. In view of this, we investigate the preemptive software project scheduling problem considering personality traits (PSPSP-PT), which aims at assigning employees with different skills and personality traits to suitable tasks and determining the start time of each task with the consideration of preemption, multi-skill, and precedence relations such that the total cost of the project is minimized. We formulate a mixed-integer linear programming model for the PSPSP-PT and design a dual priority rules-based heuristic (DPRH). Based on a benchmark dataset that is generated using a full factorial design, computational experiments are conducted to analyze the performance of our DPRH. Comparative analysis with existing algorithms reveals that our DPRH is competitive in terms of efficiency and effectiveness.

A novel bi-objective model for sustainable and efficient airport logistics management: A case study on Copenhagen airport

Due to the growing volume of air transportation and the scarcity of airport resources, aviation planners are paying more attention to the efficiency of airport resource management. Optimizing the assignment and scheduling of ground facilities plays an important role in reducing ground congestion, flight delays, and passenger dissatisfaction. To prevent any collisions among aircraft on the apron, taxi network, and runways, safety considerations should be strictly followed by traffic controllers. Minimizing taxi time is another matter of importance in airport logistics management that helps mitigate environmental concerns (e.g., fuel consumption, tire abrasion, and noise pollution). In this study, a bi-objective mixed-integer linear programming model is developed to optimize the ground resource scheduling problem, taking into account the challenges of gate assignment, taxi planning, and runway scheduling. The model is developed in a way that includes safety and sustainability considerations. The validation of the proposed model is investigated through several examples based on Copenhagen airport using the augmented ε-constraint method. Since the simultaneous scheduling of multiple resources is an NP-hard problem, two multi-objective metaheuristic algorithms are employed to deal with the large-scale problems. The performance of the algorithms is then compared in terms of coverage and convergence rates. Additionally, the results are discussed in detail to provide managerial implications for airport authorities.