Mathematical Programming Solver Research Articles

AI-Enabled Supply Chain Optimization

As global supply chains become more intricate, the significance of supply chain risk management has significantly increased. This research delves into the utilization of artificial intelligence (AI) in managing supply chain risks, analyzing cutting-edge advancements, obstacles, and potential areas for further exploration. Through a comprehensive review of literature sources like Google Scholar, Web of Science, EI, and Scopus, this study investigates how AI methods such as machine learning, deep learning, neural networks, fuzzy logic, genetic algorithms, and evolutionary algorithms can help mitigate supply chain risks. These AI technologies have demonstrated remarkable efficacy in mitigating various risks, including forecasting, anomaly detection, image recognition, text mining, logistics optimization, and emergency response tactics. Apart from AI-driven approaches, optimization solvers and algorithms play a critical role in tackling complex supply chain dilemmas. Mathematical programming solvers, including linear programming (LP), mixed-integer programming (MIP), and quadratic programming (QP), are commonly used to model and optimize supply chain networks by considering factors like cost, capacity, and demand fluctuations. Fine-tuning solver parameters and strategies to enhance computational efficiency—known as solver tuning—has been crucial in enhancing solution quality and decreasing computation time for large-scale supply chain issues. Heuristic solvers like genetic algorithms, simulated annealing, and ant colony optimization are frequently employed to resolve conflicts in supply chains. These solvers offer practical solutions to problems where exact methods are computationally impractical, allowing for swift responses to disruptions such as production delays or spikes in demand. Pyramid classification, a hierarchical approach, further refines decision-making by categorizing risks and aligning response strategies based on priority and severity. Furthermore, the integration of AI technologies and solvers enables advanced conflict resolution techniques, including scenario-based modeling and multi-objective optimization. These approaches enable decision-makers to weigh trade-offs between conflicting objectives like minimizing costs versus maximizing service levels in real-time. The study underscores that AI technologies and optimization solvers significantly bolster risk management in supply chains. Nonetheless, challenges such as data privacy concerns, security vulnerabilities, technical intricacies, and implementation obstacles pose critical barriers to widespread adoption. The study offers practical suggestions for businesses and decision-makers while pinpointing key areas for future exploration, such as devising hybrid models that merge heuristic solvers with AI for adaptive and scalable risk management strategies.

Packing squares independently

Given a set of squares and a strip with bounded width and infinite height, we consider a square strip packaging problem, which we call the square independent packing problem (SIPP), to minimize the strip height so that all the squares are packed into independent cells separated by horizontal and vertical partitions. For the SIPP, we first investigate efficient solution representations and propose a compact representation that reduces the search space from Ω(n!) to O(2n), with n the number of given squares, while guaranteeing that there exists a solution representation that corresponds to an optimal solution. Based on the solution representation, we show that the problem is NP-hard. To solve the SIPP, we propose a dynamic programming method that can be extended to a fully polynomial-time approximation scheme (FPTAS). We also propose three mathematical programming formulations based on different solution representations and confirm their performance through computational experiments with a mathematical programming solver. Finally, we discuss several extensions that are relevant to practical applications.

A choice-based approach to dynamic capacitated multi-item lot sizing with demand uncertainty

With the purpose of planning and implementing pricing decisions on a tactical level as well as production decisions on an operational level, we consider – in an integrated form – the capacitated multi-item lot sizing problem with uncertain item demands and price-dependent discrete choice demand. The model is embedded into an overarching rolling horizon procedure allowing for adaptations to changes in demand and cost parameters. We first formulate the static problem version as a nonlinear mathematical program with underlying multinomial logit demand and subsequently linearize it to make it viable for mathematical programming solvers. Uncertainty of demands is taken into account by Monte Carlo simulation. More specifically, we generate random demand scenarios and utilize them as input data for the sample average approximation problem version. We further endow the problem setting with possibilities to incorporate pricing policy requirements such as restricting the number of price adaptations or defining periods without price adaptations. Overall, the developed approach yields a powerful tool for balancing item demands via pricing in a way favorable for adhering to available production capacities and thereby striking a balance between revenues and costs. Computational results confirm that adapting prices to time-dependent demand and cost parameters is exploited effectively to maintain a deliberately controlled production environment. Moreover, the integrated pricing and production setting allows to study the effect of pricing policy restrictions and demand uncertainties upon attainable profits.

Two-set inequalities for the binary knapsack polyhedra

This paper presents two-set inequalities, a class of valid inequalities for knapsack and multiple knapsack problems. Two-set inequalities are generated from two arbitrary sets of variables from a knapsack constraint. This class of cutting planes is not a traditional type of lifting since a valid inequality over a restricted space is not required to start. Furthermore, they cannot be derived using any existing lifting technique. The paper presents a quadratic algorithm to efficiently generate many two-set inequalities. Conditions for facet-defining two-set inequalities are also derived. Computational experiments tested these inequalities as pre-processing cuts versus CPLEX, a high-performance mathematical programming solver, at default settings. Overall, two-set inequalities reduced the time to solve some benchmark multiple knapsack instances to up to 80%. Computational results also showed the potential of this new class of cutting planes to solve computationally challenging binary integer programs.

Minimizing cache usage with fixed-priority and earliest deadline first scheduling

Cache partitioning is a technique to reduce interference among tasks running on the processors with shared caches. To make this technique effective, cache segments should be allocated to tasks that will benefit the most from having their data and instructions stored in the cache. The requests for cached data and instructions can be retrieved faster from the cache memory instead of fetching them from the main memory, thereby reducing overall execution time. The existing partitioning schemes for real-time systems divide the available cache among the tasks to guarantee their schedulability as the sole and primary optimization criterion. However, it is also preferable, particularly in systems with power constraints or mixed criticalities where low- and high-criticality workloads are executing alongside, to reduce the total cache usage for real-time tasks. Cache minimization as part of design space exploration can also help in achieving optimal system performance and resource utilization in embedded systems. In this paper, we develop optimization algorithms for cache partitioning that, besides ensuring schedulability, also minimize cache usage. We consider both preemptive and non-preemptive scheduling policies on single-processor systems with fixed- and dynamic-priority scheduling algorithms (Rate Monotonic (RM) and Earliest Deadline First (EDF), respectively). For preemptive scheduling, we formulate the problem as an integer quadratically constrained program and propose an efficient heuristic achieving near-optimal solutions. For non-preemptive scheduling, we combine linear and binary search techniques with different fixed-priority schedulability tests and Quick Processor-demand Analysis (QPA) for EDF. Our experiments based on synthetic task sets with parameters from real-world embedded applications show that the proposed heuristic: (i) achieves an average optimality gap of 0.79% within 0.1× run time of a mathematical programming solver and (ii) reduces average cache usage by 39.15% compared to existing cache partitioning approaches. Besides, we find that for large task sets with high utilization, non-preemptive scheduling can use less cache than preemptive to guarantee schedulability.

A generalized Benders decomposition approach for the optimal design of a local multi-energy system

A local multi-energy system (LMES) is a decentralized energy system producing energy under multiple forms to satisfy the energy demand of a set of buildings located in its neighborhood. We study the problem of optimally designing an LMES over a multi-phase horizon. This problem is formulated as a large-size mixed-integer linear program with a block-decomposable structure involving mixed-integer sub-problems. We propose a new way to adapt a recently published framework for generalized Benders decomposition to our problem. This is done by exploiting the fact that the constraint matrix appearing in front of the first-stage variables in the coupling constraints is non-negative. The obtained generalized Benders decomposition algorithm relies on the use of a new type of non-convex Benders cuts involving indicator functions. We first prove that, under the assumption that all first-stage decision variables are integer and bounded, the finite and optimal convergence of our algorithm is guaranteed in theory. We then investigate how to obtain a good numerical performance in practice. Finally, we report the results of a computational study carried out on a real-life case study. These results show that the proposed algorithm clearly outperforms both a mathematical programming solver directly solving the problem as a whole and a state-of-the art hierarchical decomposition algorithm.

An ensemble of brain storm optimization and Q-learning methods for distributed flexible job shop scheduling problems with distribution operations

Distributed manufacturing scheduling problems have attracted much concern from both industrial and academic areas. Nevertheless, distributed scheduling problems with distribution operations are seldom studied. This work proposes a distributed flexible job shop scheduling problem with distribution operations. A set of jobs is handled at distributed flexible job shops, and then the finished jobs are transported to their corresponding customers following given due dates. First, a mixed integer programming model is established to minimize total tardiness. Second, an ensemble of brain storm optimization and Q-learning methods is developed to solve the formulated model. Six heuristics are hybridized to generate a high-quality initial population. A Q-learning method is devised by fully employing found search information to guide subsequent search processes instead of using fixed parameters as basic brain storm optimization. A variable neighborhood search method combining problem-specific knowledge is designed to further refine the found best individual. At last, the formulated model and method are compared with three state-of-the-art metaheuristics and a mathematical programming solver CPLEX via using a group of problem instances. The results and analysis demonstrate that the developed model and algorithm have more powerful competitiveness in addressing the studied problem.

A fix-and-optimize heuristic for the Unrelated Parallel Machine Scheduling Problem

This paper proposes and evaluates a matheuristic approach for the Unrelated Parallel Machine Scheduling Problem (UPMSP). The UPMSP consists of assigning jobs to unrelated parallel machines considering different processing times for the same job in different machines. Additionally, a setup time is considered between the execution of jobs in the same machine. The problem is addressed by a fix-and-optimize matheuristic that iteratively selects a subset of variables to be fixed to their current values so that the remaining variables will compose a subproblem to be optimized by a mathematical programming solver. In the proposed approach, each subproblem consists of a subset of jobs that are assigned to a subset of machines in the incumbent solution. The subproblems are solved by the state-of-the-art exact algorithm for the UPMSP. In the experiments conducted on benchmark instances, the proposed fix-and-optimize algorithm achieved remarkable results. It outperformed the standalone exact algorithm by a large margin and resulted in competitive solutions when compared to the literature’s best-performing heuristic method for the problem. The proposed algorithm obtained the best solution for 669 out of the 1000 instances addressed in this work. Among them, 338 are new best-known solutions. In general, the proposed approach excels at solving instances with a high number of jobs per machine — it resulted in the best solution for 89% of the instances with a ratio of 10 or more jobs per machine in total.

Smart City Facility Location Recommendation Algorithm Using Multimedia Data and Improved Generative Adversarial Networks

Internet of Things, low energy consumption, and intelligent and functionally integrated urban infrastructure construction are crucial elements in the development of smart cities. Distributed generation (DG) and electric vehicle charging infrastructure play a vital role in the planning and construction of smart cities. However, the uncertainty associated with the power output of distributed generation significantly impacts the planning of distribution networks. To address this issue, this paper proposes a site-selection recommendation algorithm that leverages urban multimedia data and improved generative adversarial networks. The proposed algorithm begins by modeling the uncertainty of wind power and photovoltaic (PV) generation using an enhanced conditional generative adversarial network model. To generate multimedia datasets with time-series characteristics for wind power and PV generation scenarios, monthly multimedia data labels are incorporated into the model. These multimedia datasets, representing a wide range of scenarios, are then clustered using the K-means clustering method. Furthermore, a distributed generation planning model is established, aiming to minimize the annual integrated cost. The planning problem is efficiently solved using CPLEX, a mathematical programming solver. In the simulation experiments, the proposed scheme is compared with alternative schemes. The results demonstrate that the proposed scheme achieves a significant total cost saving of 21.95% compared to the comparison scheme. Moreover, the experimental comparison reveals that the proposed scheme exhibits higher stability. Additionally, in terms of algorithm efficiency, the proposed algorithm outperforms the other three algorithms tested in terms of the number of iterations and speed. The experimental results highlight the effectiveness of the proposed planning model in improving the economy and stability of the distribution network. Furthermore, it enhances the computational efficiency of the planning problem associated with distributed power supply and electric vehicle charging stations. The findings of this research hold substantial research significance for the site selection planning of distributed power supply.

A knowledge-based multi-objective evolutionary algorithm for solving home health care routing and scheduling problems with multiple centers

Nowadays, population aging is becoming more and more serious. The elderly occupies a great many of public medical resources, which brings great pressure to the public and governments. Home health care (HHC) is treated as an alternative answer to hospitalization, and plays positive roles in alleviating the stress from the shortage of medical resources incurred by the population aging. Hence, how to effectively manage and organize the operations of HHCs becomes an all-important problem in reality. HHC routing and scheduling problems attract a great deal of interest from modeling and optimization areas. Nevertheless, most of the existing studies just focus on the HHC routing and scheduling problems with a single HHC center. This study focuses on a multi-objective HHC routing and scheduling problem with multiple centers for minimizing total service cost and total tardiness incurred by delay service while meeting caregivers’ workload and resource constraints. To cope with it, we establish a mixed integer programming model to formulate the concerned problem. Then, a knowledge-based multi-objective evolutionary algorithm (KMoEA) is specially designed by employing two local search operators with the properties of the problem. Via comparing KMoEA with five algorithms and a mathematical programming solver (CPLEX), we validate that KMoEA is an effective approach for solving the studied problem.

On good encodings for quantum annealer and digital optimization solvers

Several optimization solvers inspired by quantum annealing have been recently developed, either running on actual quantum hardware or simulating it on traditional digital computers. Industry and academics look at their potential in solving hard combinatorial optimization problems. Formally, they provide heuristic solutions for Ising models, which are equivalent to quadratic unconstrained binary optimization (QUBO). Constraints on solutions feasibility need to be properly encoded. We experiment on different ways of performing such an encoding. As benchmark we consider the cardinality constrained quadratic knapsack problem (CQKP), a minimal extension of QUBO with one inequality and one equality constraint. We consider different strategies of constraints penalization and variables encoding. We compare three QUBO solvers: quantum annealing on quantum hardware (D-Wave Advantage), probabilistic algorithms on digital hardware and mathematical programming solvers. We analyze their QUBO resolution quality and time, and the persistence values extracted in the quantum annealing sampling process. Our results show that a linear penalization of CQKP inequality improves current best practice. Furthermore, using such a linear penalization, persistence values produced by quantum hardware in a generic way allow to match a specific CQKP metric from literature. They are therefore suitable for general purpose variable fixing in core algorithms for combinatorial optimization.

An improved NSGA-II with local search for multi-objective integrated production and inventory scheduling problem

In the context of collaborative manufacturing, integrated optimization of spare parts production and inventory management is practically important. This paper investigates an integrated production and inventory scheduling (IPIS) problem based on condition-based maintenance. In respect to this problem, whereby inventory and direct supply decisions are made simultaneously to achieve a better reduction in total inventory holding costs, total tardiness and total makespan, a multi-objective IPIS model is developed. An improved non-dominated sorting genetic algorithm-II with local search (INSGA-II_LS) is proposed for the multi-objective IPIS model. In INSGA-II_LS, the encoding and population initialization suited for IPIS are designed. The detailed presentation of operators of crossover, mutation, and local search that designed for the proposed IPIS problem then follows. The mathematical programming solver CPLEX and three multi-objective evolutionary algorithms called SPEA2, PESA-II, MOEA/D are designed for comparisons against INSGA-II_LS. Experimental results show the superiority of the proposed INSGA-II_LS for the IPIS problem with respect to various multi-objective performance metrics, especially for large-scale instances.

Tabu search for proactive project scheduling problem with flexible resources

In this paper, we address a proactive resource-constrained project scheduling problem where resources have several skills and can switch the selected skill at discrete time instants. The objective of this paper is to simultaneously determine the starting times of activities and select the skills of resources to maximize the schedule robustness, while satisfying constraints of precedence, renewable resources and a project deadline. After defining the problem, an integer linear programming model is constructed and the studied problem is proven to be NP-hard. Considering the NP-hardness of the problem, we develop a tabu search embedded with two special algorithms to solve the problem. Based on the characteristics of the problem, a resource allocation algorithm for determining the skill for resources is proposed. The two special algorithms are regarded as two improvement measures to combine with the basic tabu search, leading to three versions of the tabu search, i.e., TS, TS-IM1, and TS-IM12. A commercial mathematical programming solver, which can optimally solve the model, is used as a benchmark for comparing the developed algorithm. Through a computational experiment performed on a randomly generated dataset, the effectiveness of the designed algorithms and the two improvement measures is verified, and the results for single-skilled resources as well as the skill switching of flexible resources are discussed. In addition, the effects of the key parameters on the objective function are also analysed. From the computational results, we can draw several conclusions. First, among the three developed algorithms, TS-IM12, which is the tabu search embedded with two improvement measures, performs best, and compared with the commercial software, TS-IM12 shows a higher efficiency in solving the studied problem. Second, the increase in the resource flexibility should be moderate. Third, the robustness of the baseline schedule increases with an increase of the resource flexibility, the resource availability and the project deadline. Finally, the improvement of the schedule robustness brought by the increase of the resource flexibility or the resource availability will be further magnified as the project deadline extends, while the improvement of the schedule robustness caused by the increase of the resource flexibility will be reduced as the resource availability augments.

Solving the Traveling Salesman Problem with release dates via branch and cut

In this paper we study the Traveling Salesman Problem with release dates (TSP-rd) and completion time minimization. The TSP-rd considers a single vehicle and a set of customers that must be served exactly once with goods that arrive to the depot over time, during the planning horizon. The time at which each requested good arrives is called release date and it is known in advance. The vehicle can perform multiple routes, however, it cannot depart to serve a customer before the associated release date. Thus, the release date of the customers in each route must not be greater than the starting time of the route. The objective is to determine a set of routes for the vehicle, starting and ending at the depot, where the completion time needed to serve all customers is minimized. We propose a new Integer Linear Programming model and develop a branch and cut algorithm with tailored enhancements to improve its performance. The algorithm proved to be able to significantly reduce the computation times when compared to a compact formulation tackled using a commercial mathematical programming solver, obtaining 24 new optimal solutions on benchmark instances with up to 30 customers within one hour. We further extend the benchmark to instances with up to 50 customers where the algorithm proved to be efficient. Building upon these results, the proposed model is adapted to new TSP-rd variants (Capacitated and Prize-Collecting TSP), with different objectives: completion time minimization and traveling distance minimization. To the best of our knowledge, our work is the first in-depth study to report extensive results for the TSP-rd through a branch and cut, establishing a baseline and providing insights for future approaches. Overall, the approach proved to be very effective and gives a flexible framework for several variants, opening the discussion about formulations, algorithms and new benchmark instances.

Real-time collaborative feeder vehicle routing problem with flexible time windows

Today, transportation has become a vital service due to increased competition among businesses. In this regard, customer service is one of the major challenges of logistics units. Companies can improve logistics costs, capacity utilization, service levels, and customer satisfaction by sharing new demands. The need for fast, flexible, reliable, and low-cost delivery is another challenge in urban areas for the distribution of goods. As demand increases, more vehicles are needed to deliver goods and congestion increases in urban transportation networks. A feeder vehicle routing problem (FVRP) consists of a heterogeneous fleet of vehicles, including trucks and motorcycles. In this problem, motorcycles pass easily in crowded areas, and the traffic of urban logistics is distributed easily. More importantly, it reduces the number of returns to the physical depot, lowers the costs, and leads to time savings. This study introduces the real-time collaborative feeder vehicle routing problem (RTCFVRP) with flexible time windows in which vehicles are permitted to serve customers before and after the time window. We propose the mixed-integer linear programming model with the CPLEX mathematical programming solver applying the augmented epsilon constraint. Also, multi-objective particle swarm optimization (MOPSO) and MOPSO-variable neighborhood search (MOPSO-VNS) were developed regarding the complexity of the problem. The performance of these algorithms was compared with Pareto solutions generated by the non-dominated sorting genetic algorithm-II (NSGA-II) and multi-objective evolutionary algorithm based on decomposition (MOEA/D) in both static and dynamic modes. Finally, in addition to statistical analysis, the AHP-TOPSIS method is employed to analyze and prioritize algorithms. The obtained results show the better performance of the proposed algorithm in both static and dynamic modes in small and large-size instances.

A VNS-Based Matheuristic to Solve the Districting Problem in Bicycle-Sharing Systems

A matheuristic approach that combines a reduced variable neighbourhood search (rVNS) algorithm and a mathematical programming (MP) solver to solve a novel model for the districting problem in a public bicycle-sharing system is presented. The problem is modelled as an integer programming problem. While the rVNS algorithm aims to find a high-quality set of centres for the repositioning zones, the MP solver computes the optimal allocation network of the stations to the centres of the repositioning zones. We use a predefined grid to reduce the search space the rVNS needs to explore. The proposed approach obtains promising results for small and medium-sized instances, and is also able to handle large-sized models.

A biased random‐key genetic algorithm for the home health care problem

AbstractHome health care problems consist in scheduling visits to home patients by health professionals while following a series of requirements. This paper studies the Home Health Care Routing and Scheduling Problem, which comprises a multi‐attribute vehicle routing problem with soft time windows. Additional route inter‐dependency constraints apply for patients requesting multiple visits, either by simultaneous visits or visits with precedence. We apply a mathematical programming solver to obtain lower bounds for the problem. We also propose a biased random‐key genetic algorithm, and we study the effects of additional state‐of‐the‐art components recently proposed in the literature for this genetic algorithm. We perform computational experiment using a publicly available benchmark dataset. Regarding the previous local search‐based methods, we find results up to 26.1% better than those of the literature. We find improvements from around 0.4% to 6.36% compared to previous results from a similar genetic algorithm.

A mixed-integer linear programming formulation for the modular layout of three-dimensional connected systems

Given the considerable complexity of process plants, there has been a great deal of research focused on aiding the design of plant layout through mathematical optimisation, i.e. optimising the positioning of the equipment in the plant for space and cost efficiency. Recently, the use of modular approaches within the construction industry, whereby work is performed off-site before being assembled on-site, has become a popular and powerful way of reducing build schedules and costs. Modular approaches have many other real applications where items must be packed to minimise the connections between them (e.g. piping, wiring, modular office and factory layouts) and consider the modular layout of the system.In this paper, we provide a formulation of the problem that, in addition to the standard layout problem, considers a modular block layout to allow modular construction and transportation of the plant. The problem is represented as a directed network, with the aim to pack the items into predefined containers and minimise the rectilinear distance between the connected items. We propose mixed-integer linear programming (MILP) models for the 2-dimensional and 3-dimensional problems and solve them using the state-of-the-art mathematical programming solver, Gurobi. Because of the combinatorial nature of the problem, solutions involving a large number of items may not converge and a suboptimal solution must be considered. However, our results suggest that even in the case of optimising a large number of items, the suboptimal solutions found after a reasonable number of iterations where deemed, by a domain expert, to be a good enough starting point to continue the design process, especially in the early concept phase.

A combinatorial cut-and-lift procedure with an application to 0–1 second-order conic programming

Cut generation and lifting are key components for the performance of state-of-the-art mathematical programming solvers. This work proposes a new general cut-and-lift procedure that exploits the combinatorial structure of 0–1 problems via a binary decision diagram (BDD) encoding of their constraints. We present a general framework that can be applied to a wide range of binary optimization problems and show its applicability for second-order conic inequalities. We identify conditions for which our lifted inequalities are facet-defining and derive a new BDD-based cut generation linear program. Such a model serves as a basis for a max-flow combinatorial algorithm over the BDD that can be applied to derive valid cuts more efficiently. Our numerical results show encouraging performance when incorporated into a state-of-the-art mathematical programming solver, significantly reducing the root node gap, increasing the number of problems solved, and reducing the run-time by a factor of three on average.

On the performance of task-oriented branch-and-bound algorithms for workload smoothing in simple assembly line balancing

Smoothing the workloads among the stations of an already installed assembly line is one of the major objectives in assembly line (re-)balancing. In order to find a feasible task-station assignment that distributes the total workload as equal as possible, two exact task-oriented branch-and-bound algorithms have recently been proposed. In this paper, we systematically analyse their effectiveness in solving the workload smoothing problem on simple assembly lines. In our experiments, we also examine the performance of a state-of-the-art mathematical programming solver and a ‘combined’ exact branch-and-bound procedure that integrates components of the two algorithms from the literature. In terms of theory, we show the equivalence of two recently developed local lower bounding arguments and suggest a slight improvement of the bound. We also propose an enhanced feasibility test.