Integer Linear Programming Formulation Research Articles

Mathematical programming models for order picking in warehouses with decoupling of picker and cart

As a special Travelling Salesman Problem (TSP), the Single-Picker Routing Problem (SPRP) in warehouses is of important theoretical and practical significance. In manual order picking, the items are usually picked up by a picker with a cart. To speed up the picking process, the picker is allowed to stop the cart to pick up the items individually and return to the location of the cart. This paper proposes a mixed integer linear programming formulation for an order picking problem with the decoupling of the picker and the cart in a multi-block warehouse, where the picker capacity is at most four. And based on this model, order batching problem is also analysed. Through numerical experiments, the cost of above decoupling problems is computed under different settings of the speed ratio and the capacity of the picker while he/she is travelling alone. The results indicate how the decoupling of picker and cart leads to cost reduction. Our model shows good performance for orders of small and medium scale and hence has great potential to improve the order picking operation which is executed by decoupling containers and robots/vehicles in modern digital warehouse systems.

Optimizing modular equipment in the design of multi-product reconfigurable production lines

This work addresses the problem of the design and balancing of a modular multi-product line in a reconfigurable environment. The study emphasizes the importance of modularity in the reconfigurable manufacturing systems (RMS), which allows fast, efficient, and cost-effective adaptation to changes by rearranging, moving, or removing modules. The line consists of a fixed number of modular machines arranged linearly and connected by a material handling system. Each machine has a limited number of slots for modules. The line can handle several product types per batch, and each batch requires a specific line configuration to be produced. The objective consists in identifying the minimum number of modules needed and their arrangement in different configurations to produce each product type. To solve this problem, a compact integer linear programming (ILP) formulation is initially investigated. Then an ILP reformulation, based on a decomposition approach, as well as a heuristic that uses this ILP reformulation are developed. The results demonstrate the effectiveness of the proposed methods.

Cost minimization of multi-class demands groomed over multi-rate OTN interfaces considering WDM-layer constraints

This paper introduces a practical comprehensive approach to optimize the cost of installed Optical Transport Network (OTN) interfaces, as well as the required number of wavelengths in the substrate Wavelength Division Multiplexing (WDM) optical network. The approach assumes grooming variable-bit-rate traffic demands on multi-rate OTN interfaces. We propose an Integer Linear Programming (ILP) formulation to address the combined problem of OTN interface deployment, virtual-network formation, traffic grooming on the established distinct-capacity virtual paths, and virtual network embedding on the WDM physical substrate. The presented model focuses on mitigating the capital expenditure required to deploy OTN interfaces, usually the most costly element in the network, as well as minimizing the required number of wavelengths in the Routing and Wavelength Assignment (RWA) problem, contributing to the overall reduction in network costs and ensuring open capacity for future expansions. As the RWA problem is NP-hard and involves significant computational costs, we also propose a multi-step heuristic approach to tackle larger network instances more efficiently. In this approach, the optimization steps for the OTN design and the RWA solutions are conducted independently. For the OTN-layer planning, the optimum ILP formulation (OTN-ILP) is maintained, so that the optimum solution for the cost of installed interfaces on the OTN layer is guaranteed. The RWA problem is proposed through either an ILP formulation (RWA-ILP) or a heuristic approach. The RWA-ILP model can find the same or slightly superior number of wavelengths compared to the optimum solution of the Integrated ILP formulation, but it can provide results in reasonable times only for small networks. On the other hand, the heuristic RWA is able to run any network size, but it requires more wavelengths compared to the RWA-ILP model.

Integer Linear Programming Models and Greedy Heuristic for the Minimum Weighted Independent Dominating Set Problem

This paper explores the Minimum Weighted Independent Dominating Set Problem and proposes novel approaches to tackle it. Namely, two integer linear programming formulations and a fast greedy heuristic as an alternative approach are proposed. Extensive computational experiments are conducted to evaluate the performance of these approaches on the established set of benchmark instances for the problem. The obtained results demonstrate that the introduced integer linear programming models are able to achieve optimal solutions on all instances with 100 nodes and significantly outperform existing exact methods on numerous other instances. Additionally, the greedy heuristic exhibits superior performance compared to competing greedy heuristics, particularly on random graphs. These findings suggest promising directions for future research, including the integration of these methods into hybrid algorithms or metaheuristic frameworks.

A logistics provider’s profit maximization facility location problem with random utility maximizing followers

We introduce a strategic decision-making problem faced by logistics providers (LPs) seeking facility location decisions that lead to profitable operations. The profitability depends on the revenue generated through agreements with shippers, and the costs arising when satisfying these agreements. The latter depends, in turn, on the service levels and characteristics of the shippers’ customers. However, at a strategic level, the LP has imperfect information thereof.We propose a stochastic bilevel formulation where a given LP (leader) anticipates the decisions of shippers (followers) arising from a random utility maximization model. Using a sample average approximation and properties of the associated optimal solutions, we propose a heuristic that can compute high-quality solutions. It is based on a reduced single-level mixed integer linear programming formulation that can be solved by a general-purpose solver after a preprocessing phase. We can quickly identify situations that lead to zero expected profit for the LP. Experimental results show that expected profit highly depends on shippers’ price sensitivities. Underestimating price sensitivities can lead to an overestimation of the expected profit.

Optimizing service networks to support freight rail decarbonization: Flow selection, facility location, and energy sourcing

We present a framework to support decarbonization of energy intensive transportation systems offering periodic service on expansive networks (e.g., freight rail, trucking, and intercity bus services). The framework consists of two optimization problems that respectivelyaddress (i) flow selection and facility location, and (ii) energy sourcing/procurement at the service facilities to enable the selected flows. The framework generalizes mixed integer linear programming formulations for flow refueling facility location and flow-based set cover models appearing in the literature to situations where it is of interest to account for repositioning of assets along cyclical trajectories to allow for periodic service, and to account for intermediate flow capture (i.e., trip chaining). The framework also consists of a minimum cost network flow model to determine optimal energy sourcing and distribution strategies, which dictate capacity requirements at the service facilities. The energy demands are obtained from the solution to the flow selection and facility location model. To illustrate the framework, we analyze the deployment of charging stations to support battery-electric locomotive service on a subset of the US freight rail network (i.e., an aggregate network of 3 Class I Railroads). The results show that the deployment of 30 charging stations can support battery-electric locomotives (with 1600-km ranges) to serve 86% of distance-weighted flows (ton-km) and reduce emissions by approximately 50%.

Intelligent Resource Orchestration for 5G Edge Infrastructures

The adoption of edge infrastructure in 5G environments stands out as a transformative technology aimed at meeting the increasing demands of latency-sensitive and data-intensive applications. This research paper presents a comprehensive study on the intelligent orchestration of 5G edge computing infrastructures. The proposed Smart 5G Edge-Cloud Management Architecture, built upon an OpenNebula foundation, incorporates a ONEedge5G experimental component, which offers intelligent workload forecasting and infrastructure orchestration and automation capabilities, for optimal allocation of virtual resources across diverse edge locations. The research evaluated different forecasting models, based both on traditional statistical techniques and machine learning techniques, comparing their accuracy in CPU usage prediction for a dataset of virtual machines (VMs). Additionally, an integer linear programming formulation was proposed to solve the optimization problem of mapping VMs to physical servers in distributed edge infrastructure. Different optimization criteria such as minimizing server usage, load balancing, and reducing latency violations were considered, along with mapping constraints. Comprehensive tests and experiments were conducted to evaluate the efficacy of the proposed architecture.

Coordinating scheduling and rejection decisions in a two-machine flow shop scheduling problem

We study a two-machine flow shop scheduling problem where any operation can be rejected at a certain cost. A solution for such a problem requires two sets of decisions. The first involves the partition of the set of operations into two subsets: the set of operations that are accepted for scheduling in the shop, and the set of rejected operations. The second decision involves scheduling the set of accepted operations in the shop. The objective is to find a solution that minimizes the sum of the makespan and the total rejection cost. We prove that the problem is NP-hard even if all processing operations have identical processing times and identical rejection costs on either one of the two machines. We show, however, that the problem is fixed parameterized tractable with respect to a parameter that combine the number of different processing times on both machines with the number of different rejection costs on one out of the two machines. We also provide a pseudo-polynomial time algorithm for the problem, which we then convert into a fully polynomial time approximation scheme. This is achieved by dividing the problem into a set of subproblems and deriving a fully polynomial time approximation scheme for each one of them, separately. Finally, we present an integer linear programming formulation of the problem and two simple 2-approximation algorithms.

Solving the Set Covering Problem with Conflicts on Sets: A new parallel GRASP

In this paper, we analyze a new variant of the well-known NP-hard Set Covering Problem, characterized by pairwise conflicts among subsets of items. Two subsets in conflict can belong to a solution provided that a positive penalty is paid. The problem looks for the optimal collection of subsets representing a cover and minimizing the sum of covering and penalty costs. We introduce two integer linear programming formulations and a quadratic one for the problem and provide a parallel GRASP (Greedy Randomized Adaptive Search Procedure) that, during parallel executions of the same basic procedure, shares information among threads. We tailor such a parallel processing to address the specific problem in an innovative way that allows us to prevent redundant computations in different threads, ultimately saving time. To evaluate the performance of our algorithm, we conduct extensive experiments on a large set of new instances obtained by adapting existing instances for the Set Covering Problem. Computational results show that the proposed approach is extremely effective and efficient providing better results than Gurobi (tackling three alternative mathematical formulations of the problem) in less than 1/6 of the computational time.

Performance Optimization Across the Edge-Cloud Continuum: A Multi-agent Rollout Approach for Cloud-Native Application Workload Placement

The advancements in virtualization technologies and distributed computing infrastructures have sparked the development of cloud-native applications. This is grounded in the breakdown of a monolithic application into smaller, loosely connected components, often referred to as microservices, enabling enhancements in the application’s performance, flexibility, and resilience, along with better resource utilization. When optimizing the performance of cloud-native applications, specific demands arise in terms of application latency and communication delays between microservices that are not taken into consideration by generic orchestration algorithms. In this work, we propose mechanisms for automating the allocation of computing resources to optimize the service delivery of cloud-native applications over the edge-cloud continuum. We initially introduce the problem’s Mixed Integer Linear Programming (MILP) formulation. Given the potentially overwhelming execution time for real-sized problems, we propose a greedy algorithm, which allocates resources sequentially in a best-fit manner. To further improve the performance, we introduce a multi-agent rollout mechanism that evaluates the immediate effect of decisions but also leverages the underlying greedy heuristic to simulate the decisions anticipated from other agents, encapsulating this in a Reinforcement Learning framework. This approach allows us to effectively manage the performance–execution time trade-off and enhance performance by controlling the exploration of the Rollout mechanism. This flexibility ensures that the system remains adaptive to varied scenarios, making the most of the available computational resources while still ensuring high-quality decisions.

Algorithmic aspect on total Roman {2}-domination of Cartesian products of paths and cycles

A total Roman {2}-dominating function (TR2DF) on a graph G with vertex set V is a function f : V → {0, 1, 2} having the property that for every vertex v with f(v) = 0, ∑u∈N (v) f(u) ≥ 2, where N(v) represents the open neighborhood of v, and the subgraph of G induced by the set of vertices with f(v) > 0 has no isolated vertex. The weight of a TR2DF f is the value w(f) = ∑v∈V f(v), and the minimum weight of a TR2DF of G is the total Roman {2}-domination number γtR2(G). The total Roman {2}-domination problem (TR2DP) is to determine the value γtR2(G). In this paper, we first propose an integer linear programming (ILP) formulation for the TR2DP. Furthermore, we apply the discharging approach to determine the total Roman {2}-domination number for some Cartesian products of paths and cycles.

The maximum length car sequencing problem

This paper introduces the maximum length car sequencing problem to support the assembly operations of a multinational automotive company. We propose an integer linear programming (ILP) formulation to schedule the maximum number of cars without violating the so-called option constraints. In addition, we present valid combinatorial lower and upper bounds, which can be calculated in less than 0.01 s, as well as binary and iterative search algorithms to solve the problem when good primal bounds are not readily available. To quickly obtain high-quality solutions, we devise an effective iterated local search algorithm, and we use the heuristic solutions as warm start to further enhance the performance of the exact methods. Computational results demonstrate that relatively low gaps were achieved for benchmark instances within a time limit of ten minutes. We also conducted an instance space analysis to identify the features that make the problem more difficult to solve. Moreover, the instances reflecting the company’s needs could be solved to optimality in less than a second. Finally, simulations with real-world demands, divided into shifts, were conducted over a period of four months. In this case, we use the proposed ILP model in all shifts except the last one of each month, for which we employ an alternative ILP model to sequence the unscheduled cars, adjusting the pace of the assembly line in an optimal fashion. The results pointed out that the latter was necessary in only one of the months.

On maximal Roman domination in graphs: complexity and algorithms

For a simple undirected connected graph G = (V,E), a maximal Roman dominating function (MRDF) of G is a function f : V (G) → {0,1,2} with the following properties: (i) For every vertex v ∈ {v ∈ V |f(v) = 0}, there exists a vertex u ∈ N(v) such that f(u) = 2. (ii) The set {v ∈ V |f(v) = 0} is not a dominating set of G; In other words, there exists a vertex v ∈ {v ∈ V |f(v) = 0} such that N(v) ∩ {u ∈ V |f(u) = 0} = ∅. The weight of an MRDF of G is the sum of its function values over all vertices, denoted as f(G) =Σv∈V (G)f(v), and the maximal Roman domination number of G, denoted by γmR(G), is the minimum weight of an MRDF of G. In this paper, we establish some bounds of the maximal Roman domination number of graphs. Additionally, we develop an integer linear programming formulation to compute the maximal Roman domination number of any graph. Furthermore, we prove that maximal Roman domination problem (MRD) is NP-complete even restricted to star convex bipartite graphs and chordal bipartite graphs. Lastly, we show the maximal Roman domination number of threshold graphs, trees, and block graphs can be computed in linear time.

Determining cost-efficient controls of electrical energy storages using dynamic programming

Volatile electrical energy prices are a challenge and an opportunity for small and medium-sized companies in energy-intensive industries. By using electrical energy storage and/or an adaptation of production processes, companies can significantly profit from time-depending energy prices and reduce their energy costs.We consider a time-discrete optimal control problem to reach a desired final state of the energy storage at a certain time step. Thereby, the energy input is discrete since only multiples of 100 kWh can be purchased at the EPEX SPOT market. We use available price estimates to minimize the total energy cost by a rounding based dynamic programming approach. With our model non-linear energy loss functions of the storage can be considered and we obtain a significant speed-up compared to the integer (linear) programming formulation.

Virtual network function reconfiguration in 5G networks: An optimization perspective

AbstractOne of the major challenges in managing 5G networks is the reconfiguration of network slices. The task covers in particular reconfiguration and relocation of virtual network functions (VNFs) so as to match the service requirements of the slices and the availability of resources of the data centers. In this article, we study in deep the problem of optimal VNFs reconfiguration analyzing a number of its variants. We define two exact, compact integer linear programming formulations of the VNFs reconfiguration problem, and derive two approximate solution algorithms, one of them based on the column generation method. Numerical tests and comparisons illustrate the performance of the algorithms and the quality of the provided solutions.

Project acceleration using temporary workers with heterogeneous efficiency: optimal and best recruitment policies

ABSTRACT This paper investigates recruitment decisions of temporary work with several levels of experience, from beginners to experts (heterogeneous efficiency) to accelerate a manufacturing project with the objective of labour cost minimisation. A Mixed Integer Linear Program formulation of the problem is developed to find the optimal pool of temporary work of each experience level so as to minimise the cost subject to a desired deadline. The deadline is decreased decrementally in order to identify any potential patterns of recruitment along the reduction in the project duration. A variety of projects, efficiency levels with respect to years of experience and cost scenarios are considered in an experimental framework. Simulation results show a high sensitivity of the optimal pool of temporary work to the project type, efficiency levels and costs. A worst-case analysis suggests that using solely expert temporary workers might be the best recruiting policy to accelerate a project, whatever the desired deadline.

Discretization-based solution approaches for the circle packing problem

The problem of packing a set of circles into the smallest surrounding container is considered. This problem arises in different application areas such as the automobile, textile, food and chemical industries. The so-called circle packing problem can be cast as a non-convex quadratically constrained program, and is difficult to solve in general. An iterative solution approach based on a bisection-type algorithm on the radius of the larger circle is provided. The present algorithm discretizes the container into small cells and solves two different integer linear programming formulations proposed for a restricted and a relaxed version of the original problem. The present algorithm is enhanced with solution space reduction, bound tightening and variable elimination techniques. Then, a computational study is performed to evaluate the performance of the algorithm. The present algorithm is compared with the BARON and Gurobi tm optimizers, which solve the original nonlinear formulation, and heuristic methods from the literature. Promising results are obtained.

A cut-and-solve algorithm for virtual machine consolidation problem

The virtual machine consolidation problem attempts to determine which servers to be activated, how to allocate virtual machines to the activated servers, and how to migrate virtual machines among servers such that the summation of activated, allocation, and migration costs is minimized subject to the resource constraints of the servers and other practical constraints. In this paper, we first propose a new mixed integer linear programming formulation for the virtual machine consolidation problem. We show that compared with existing formulations, the proposed formulation is much more compact in terms of smaller numbers of variables or constraints, which makes it suitable for solving large-scale problems. We then develop a cut-and-solve algorithm, a tree search algorithm to efficiently solve the virtual machine consolidation problem to optimality. The proposed cut-and-solve algorithm is based on a novel relaxation of the virtual machine consolidation problem that provides a stronger lower bound than the natural continuous relaxation of the virtual machine consolidation problem, making a smaller search tree. By extensive computational experiments, we show that (i) the proposed formulation significantly outperforms existing formulations in terms of solution efficiency; (ii) compared with standard mixed integer linear programming solvers, the proposed cut-and-solve algorithm is much more efficient; and (iii) compared with an existing heuristic algorithm, the proposed cut-and-solve algorithm is better able to balance workload distribution across servers and achieve higher resource utilization.

Algorithms and computational study on a transportation system integrating public transit and ridesharing of personal vehicles

The potential of integrating public transit with ridesharing includes shorter travel time for commuters and higher occupancy rate of personal vehicles and public transit ridership. In this paper, we describe a centralized transit system that integrates public transit and ridesharing to reduce travel time for commuters. In the system, a set of ridesharing providers (drivers) and a set of public transit riders are received. The optimization goal of the system is to assign riders to drivers by arranging public transit and ridesharing combined routes for as many riders as possible subject to shorter commuting time. As an exact algorithm approach, we give a (0,1) integer linear programming (ILP) formulation based on a hypergraph representation of the problem. Leveraging the ILP and hypergraph, we give approximation algorithms based on LP-rounding and hypergraph matching/weighted set packing, respectively. As a case study, we conduct an extensive computational study based on real-world public transit dataset and ridesharing dataset in Chicago city. To evaluate the effectiveness of the transit system and our algorithms, we generate data instances from the datasets. The experimental results show that more than 60% of riders are assigned to drivers on average, riders’ commuting time is reduced by 23% and vehicle occupancy rate is improved to almost 3. Our proposed algorithms are efficient for practical scenarios.

Exact and heuristic solution approaches for the multi-objective AGV scheduling problem with battery constraints

Automated Guided Vehicles (AGVs) are self-driving vehicles used for transporting goods and materials throughout different areas, such as shipping and receiving areas, storage facilities, and workstations. The popularity of AGVs has grown rapidly due to their many benefits, including flexibility in processes, efficient use of space, product safety, and computer integration and control. The design and management of AGV-based systems present various tactical decisions, such as fleet sizing and flow path design, as well as operational decisions such as dispatching, routing, and scheduling.In this study, we focus on a scheduling problem faced by a manufacturing company that utilizes AGVs subject to battery constraints for the horizontal movement of materials. Unlike previous studies, we investigate a Multi-objective AGV scheduling problem where both the AGV fleet size and the makespan need to be minimized. Additionally, AGV charging time depends on battery depletion. To support the company decision-making system, we propose a Mixed Integer Linear Programming formulation and a genetic algorithm. We have tested and validated our proposed methods using real-world instances provided by a manufacturing company. The results show the efficacy and the scalability of the proposed solution methods.