Linear Programming Formulation Research Articles

Adjustable Robust Low-Carbon Unit Commitment With Nonanticipativity by Linear Programming

Thermal power generation is the main source of carbon emissions. Adopting low-carbon dispatching and developing renewable energy sources (RESs) are effective ways to reduce carbon emissions. This article constructs a mixed-integer programming (MIP) model of low-carbon full-scenario unit commitment with nonanticipativity. To overcome the curse of dimensionality problem caused by the use of a massive number of scenarios, we employ the adjustable robust optimization approach (AROA) to reformulate the full-scenario unit commitment as a deterministic robust model. In addition, we establish the precise adjustable robust convex hull of generation constraints in a higher dimensional space and generate a relaxed linear programming (LP) formulation of the original MIP model. Then, we design a heuristic method for obtaining a near-optimal feasible solution by converting a large-scale MIP into an LP model that can be solved in polynomial time. The numerical experiments presented in this article demonstrate the effectiveness and efficiency of the proposed method.

Utilizing Tax Mechanisms to Foster Research and Development Activities in Enterprises in Poland

This paper intends to present different models for the decision-making process in mining enterprises. In the beginning, the most significant parameters of optimization problems, along with criteria optimization models are introduced, while both quantitative and qualitative optimization criteria are identified next in the article. The paper deals then with introducing the main issues related to optimal decision models in the mining industry through linear programming and the problem of re-optimizations. The analysis continues by presenting the mathematical formulation of linear programming, giving general considerations on applicability in the mining industry, and providing an example of a linear programming problem in the mining industry. Hence, the rest of the paper is dedicated to some numerical examples, first of a model of the optimal planning of the exploitation of the panels regarding the production of copper and gold, and second, giving a numerical example of the linear programming problem in the mining industry, to demonstrate its validity in investment decision-making for mining exploitation.

Econometric Models for Optimal Decisions in Mining Industry

This paper intends to present different models for the decision-making process in mining enterprises. In the beginning, the most significant parameters of optimization problems, along with criteria optimization models are introduced, while both quantitative and qualitative optimization criteria are identified next in the article. The paper deals then with introducing the main issues related to optimal decision models in the mining industry through linear programming and the problem of re-optimizations. The analysis continues by presenting the mathematical formulation of linear programming, giving general considerations on applicability in the mining industry, and providing an example of a linear programming problem in the mining industry. Hence, the rest of the paper is dedicated to some numerical examples, first of a model of the optimal planning of the exploitation of the panels regarding the production of copper and gold, and second, giving a numerical example of the linear programming problem in the mining industry, to demonstrate its validity in investment decision-making for mining exploitation.

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.

Evaluating and extending speedup techniques for optimal crossing minimization in layered graph drawings.

A layered graph is an important category of graph in which every node is assigned to a layer, and layers are drawn as parallel or radial lines. They are commonly used to display temporal data or hierarchical graphs. Previous research has demonstrated that minimizing edge crossings is the most important criterion to consider when looking to improve the readability of such graphs. While heuristic approaches exist for crossing minimization, we are interested in optimal approaches to the problem that prioritize human readability over computational scalability. We aim to improve the usefulness and applicability of such optimal methods by understanding and improving their scalability to larger graphs. This paper categorizes and evaluates the state-of-the-art linear programming formulations for exact crossing minimization and describes nine new and existing techniques that could plausibly accelerate the optimization algorithm. Through a computational evaluation, we explore each technique's effect on calculation time and how the techniques assist or inhibit one another, allowing researchers and practitioners to adapt them to the characteristics of their graphs. Our best-performing techniques yielded a median improvement of 2.5-17× depending on the solver used, giving us the capability to create optimal layouts faster and for larger graphs. We provide an open-source implementation of our methodology in Python, where users can pick which combination of techniques to enable according to their use case. A free copy of this paper and all supplemental materials, datasets used, and source code are available at https://osf.io/5vq79.

Priority based energy and load aware routing algorithms for SDN enabled data center network

Nowadays, to accommodate the swift expansion of cloud computing, big data, and other emerging technologies, integrating data center networks and SDN (Software Defined Network) is suggested, which also improves the network management’s convenience and flexibility. As network devices in these data centers continue to increase at a high rate, efficient route management to handle these devices’ vast amounts of data has emerged as a significant concern. This work suggests a novel admission and routing scheme considering the importance or priorities of the network flows, path energy, and routing path load. Using the SDN paradigm in DCN (Data Center Network), we first develop a Mixed Integer Linear Programming (MILP) formulation by jointly considering flow priority, path energy, and path load. The MILP formulation can maximize the number of flows as well as minimize the energy consumption and load variance in the network, or it can make a trade-off among all three. However, due to the potentially long running time, later, we introduce two greedy methods, Priority Based Energy Minimization Algorithm (PEMA) and Priority Based Evenly Load Distribution Algorithm (PEDL), where PEMA aims to maximize the flow with less energy, and PEDL focuses on maximizing the flow with reduced load variation. Finally, the developed routing strategies are implemented, and the simulation results demonstrate the out-performance of our work compared to the existing works in terms of successful flow ratio, energy savings, and load balancing.

Land use planning to minimize the export of phosphorus: An optimization model for dairy production at a catchment area scale

The intensification of activities in the Santa Lucia River basin in Uruguay has resulted in an increase in the concentration of nutrients in surface waters with the consequent acceleration of the eutrophication. Dairy production, of great relevance in the area, needs to increase production to ensure the economic viability of the activity. In this context, it is necessary to find solutions that address trade-offs between productivity and polluting potential. A dairy sub-basin of the Santa Lucia River was selected as a case study to develop an optimization model with the aim of minimizing phosphorus exportation over a four-year planning horizon while meeting restrictions on the crop sequence allowed, dry matter productivity and number of crops in each sub-basin producer. An integer linear programming formulation was proposed to model the problem. It was possible to obtain a model that managed to reduce phosphorus runoff losses by planning land uses and complying the restrictions. Exists a wide field of improvement of the model, which is a useful tool for analysis of scenarios and trade-off between productivity and environmental care.

The capacitated family traveling salesperson problem

AbstractThe capacitated family traveling salesperson problem (CFTSP) is about a graph in which nodes are partitioned into disjoint and differently weighted families. The objective is to find the shortest route that visits a given number of nodes in each family with a set of capacitated agents. The CFTSP is a new variant of the family traveling salesman problem (FTSP). Nevertheless, existing exact and metaheuristic methods for the FSTP cannot be straightforwardly applied due to the feasibility condition that requires agents to stay within their capacity. Thus, we propose integer linear programming formulations with five different subtour inequalities sets that are tested and compared with the classical approaches. In addition, we propose a biased random‐key genetic algorithm with four decoder algorithms that find high‐quality solutions in short computational times. Based on our experiments, we have found that our methods are effective in addressing subtour avoidance and ensuring feasible assignments.

Joint fiber and MEC deployment for sparsely populated areas

The deployment of multi-access edge computing (MEC) networks gives rise to the MEC placement problem, which deals with finding the right server locations to reduce the cost and guarantee network performance. Multiple papers have been presented to solve this problem, but they are usually oriented to urban areas where short distances and high-quality network infrastructure are assumed. When this problem must be solved for sparsely populated areas, like rural environments, the connectivity is not always granted and the deployment of such connectivity using fiber technologies should be included in the problem. In contrast to urban areas, where the density of users is high and therefore the main problem is capacity, in sparsely populated areas, the problem lies in how to cost-effectively plan the MEC sites and the interconnecting network while meeting the delay constraints of the services offered through that network. This paper proposes a technique to solve the MEC placement problem considering the joint deployment of the optical network required to interconnect the base stations and the MEC servers. It consists of a three-phase scheme, which combines a spanning tree topology, for fiber deployment, with the use of mixed integer linear programming (MILP) formulations to minimize MEC servers and MEC data centers (MEC-DCs). We have applied the technique in a case study for a province in Spain (Valladolid, 8110km2 ), obtaining a reduction of around 50% of the total cost when compared to a previous work. In addition, a clustering method is proposed to improve the scalability of the model for large scenarios. A simulation study is also presented to demonstrate the performance of the proposal assuming a 94,226km2 region (Castilla y León) with 1576 base stations.

Modeling and optimal operation of reversible solid oxide cells considering heat recovery and mode switching dynamics in microgrids

The reversible solid oxide cell (rSOC) is a promising technology for advancing energy decarbonization by enabling bidirectional conversion between electricity and hydrogen in a single device. However, previous studies have not fully explored the operational flexibility of rSOCs due to inadequate consideration of heat recovery potentials and dynamics of operating mode transitions. To address this research gap, this paper presents a model-based optimal operation method for managing multi-energy transactions in rSOC-based microgrids, aiming to minimize operation costs. The method incorporates detailed operational models of the rSOC, including a lumped thermal model to account for heat recovery capability and modeling of various operating modes and their transitions. Additionally, a linearization process is introduced to address nonlinear and implicit operational constraints, resulting in a computationally efficient mixed-integer linear programming (MILP) formulation for the operation model. Comparative case studies are conducted using modified energy portfolios of a Danish energy island. The results demonstrate that the proposed method effectively captures operating mode transitions within the rSOC and enhances its profitability via waste heat recovery. Notably, the rSOC model contributes to enhanced operational flexibility through heat recovery behaviors and a wider temperature range, resulting in substantial economic savings for the microgrid.

A hybrid approach for unit commitment with splitting technique and local search

In the modern electricity market, solving the unit commitment (UC) problem becomes more challenging due to its large number of binary variables and coupling constraints. Therefore, the goal of this paper is to obtain nearly optimal solutions for such complex UC problems within specific time limits. To achieve this, a hybrid approach is presented in this paper, which incorporates a splitting technique and a local search for UC. The hybrid approach involves two steps. In the first step, the augmented Lagrange function (ALF) of UC is decomposed using a splitting technique based on the alternating direction method of multipliers (ADMM). The resulting sub-problems can be effectively solved in a distributed manner by utilizing Lagrange duality theory and binary variable characteristics. Additionally, the penalty parameters of the ALF are updated effectively with the sub-gradient method to diminish the influence of the initial penalty parameters on the algorithm. In the second step, the appropriate relaxation unit is selected using the binary variable characteristics in the current solution of ADMM. Further, the high-quality feasibility of the final solution is ensured through local search. The hybrid approach utilizes ADMM to quickly fix most binary variables in the UC model and then leverages the CPLEX solver to solve the mixed-integer linear programming (MILP) formulation of UC by fixing a large number of binary variables. Consequently, the CPLEX solver needs to branch fewer binary variables, reducing the local search time. The hybrid approach is applied to solve several power systems consisting of up to 1000 units. Compared with the CPLEX solver, the solution of the hybrid approach is found to be stable, and the relative gap of the solution is less than 1 %, demonstrating the effectiveness and performance of the proposed approach while meeting the time limit requirements of the power industry dispatching.

Valid inequalities for the k-Color Shortest Path Problem

Given a digraph D=(V,A) where each arc (i,j)∈A has a cost dij∈R+ and a color c(i,j), a positive integer k, and vertices s,t∈V, the k-Color Shortest Path Problem consists in finding a path from s to t of minimum cost while using at most k distinct arc colors. We propose valid inequalities for the problem that proved to strengthen the linear relaxation of an existing Integer Linear Programming formulation for the problem. One exponential set of valid inequalities defines a new formulation for the problem that is solved by using a branch-and-cut algorithm. We introduce more challenging instances for the problem and present numerical experiments for both the benchmark and the new instances. Finally, we evaluate the individual and the collective use of the valid inequalities. Computational results for the proposed ideas and for existing solution approaches for the problem showed the effectiveness of the new inequalities in handling the new instances, both in terms of execution times and improvement of the linear relaxed solutions.

LP formulation of stochastic Bayesian two‐player zero‐sum games with long horizon

AbstractThis article studies two‐player zero‐sum stochastic Bayesian games where each player has its own dynamic state that is unknown to the other. To compute the optimal strategy for the players, prior work uses sequence form to construct linear programming formulation, whose size grows exponentially with respect to the time horizon of the game. The exponential computational complexity restricts us in games with short horizon. While window‐by‐window or receding horizon methods can be used to extend the horizon, how to update the initial parameters in each window is still unknown. Based on the existing results about dual games, this article proposes LP formulations to update the initial parameters in each window. A window‐by‐window method is given with performance analysis. The main results are demonstrated in a security problem of underwater sensor networks.

Data-driven evacuation and rescue traffic optimization with rescue contraflow control

In response to local sudden disasters, e.g., high-rise office or residential building fire disasters, road occupation can cause conflicts, and traffic directions may be opposite between evacuation vehicles and rescue vehicles; moreover, lane contraflow can be adopted to meet these surge traffic demands. However, lane contraflow that provides more roads for rescue vehicles reduces the traffic supply in the evacuation direction. It is unclear how to control the number of contraflow roads used by rescue vehicles to coordinate evacuation and rescue traffic operations. Here, we adjust the critical rescue traffic volume of reversing the normal road traffic direction to control rescue contraflow. Additionally, we propose a multiobjective mixed integer linear programming formulation for evacuation and rescue traffic optimization. Additionally, considering that the upper limit of the critical rescue traffic volume is unknown and that the proposed formulation includes multiple objectives and multi-priority vehicle classes, a three-stage solving algorithm is developed. Next, a large-scale evacuation and rescue traffic optimization result dataset is obtained for the Nguyen–Dupuis road network, and the impact of different rescue contraflow control plans on evacuation and rescue traffic is studied based on data-driven statistical analysis. The results show that by adjusting the optimal rescue traffic route, the critical rescue traffic volume for reversing the normal road traffic direction can reduce the interference of rescue traffic to evacuation traffic operation performance without reducing rescue traffic operation performance, and can be used to coordinate evacuation and rescue traffic operation under rescue contraflow.

A Unified Approach for Resilience and Causal Responsibility with Integer Linear Programming (ILP) and LP Relaxations

What is a minimal set of tuples to delete from a database in order to eliminate all query answers? This problem is called "the resilience of a query" and is one of the key algorithmic problems underlying various forms of reverse data management, such as view maintenance, deletion propagation and causal responsibility. A long-open question is determining the conjunctive queries (CQs) for which resilience can be solved in PTIME. We shed new light on this problem by proposing a unified Integer Linear Programming (ILP) formulation. It is unified in that it can solve both previously studied restrictions (e.g., self-join-free CQs under set semantics that allow a PTIME solution) and new cases (all CQs under set or bag semantics). It is also unified in that all queries and all database instances are treated with the same approach, yet the algorithm is guaranteed to terminate in PTIME for all known PTIME cases. In particular, we prove that for all known easy cases, the optimal solution to our ILP is identical to a simpler Linear Programming (LP) relaxation, which implies that standard ILP solvers return the optimal solution to the original ILP in PTIME. Our approach allows us to explore new variants and obtain new complexity results. 1) It works under bag semantics, for which we give the first dichotomy results in the problem space. 2) We extend our approach to the related problem of causal responsibility and give a more fine-grained analysis of its complexity. 3) We recover easy instances for generally hard queries, including instances with read-once provenance and instances that become easy because of Functional Dependencies in the data. 4) We solve an open conjecture about a unified hardness criterion from PODS 2020 and prove the hardness of several queries of previously unknown complexity. 5) Experiments confirm that our findings accurately predict the asymptotic running times, and that our universal ILP is at times even quicker than a previously proposed dedicated flow algorithm.

Computational complexity and linear formulations for optimizing the location of shared mobility stations

AbstractProperly planned shared mobility services can be attractive even for who owns and drives a private vehicle, but would consider not owning it anymore if a cheaper, more sustainable, and accessible transport service is available. However, planning such services may become computationally intractable as demand and coverage widen, requiring different methods for optimizing them. This work contextualizes the computational challenges in defining station locations for shared mobility services; proves the NP‐hardness of optimizing the exact locations for shared mobility stations; proves the NP‐hardness of selecting street segments to have shared mobility stations; proposes mixed‐integer linear programming formulations for both problems; proposes and compares the performance of these formulations to two heuristics; and applies these formulations to the road network of the metropolitan region of São Paulo in Brazil. Results show the trade‐off between computational efforts and the precision of optimal solutions by choosing which problem to solve.

Towards a circular economy with waste‐to‐resource system optimization

AbstractMotivated by mounting pressures to achieve environmental sustainability, and the emergence of online waste exchange platforms, in this work we propose an optimization‐based framework for studying waste‐to‐resource formation in the industry. We first develop a linear programming market clearing model comprising self‐interested agents participating in waste‐to‐resource trading. We then embed this in a bi‐level capacity optimization problem under uncertainties in the agents' reserve prices, the goal of which is to maximize the economic savings achieved by waste‐to‐resource. We show that the resulting robust optimization models admit linear mixed integer programming formulations that can be readily computed using available solvers. Furthermore, we provide the formulas for applying an equitable levy and incentive scheme, to enable capacity cost‐sharing among the agents. Numerical studies based on a case of organic waste streams demonstrate the usefulness of the proposed models in generating valuable insights for decision support in the design of waste‐to‐resource markets.

Inter-array cabling optimization in offshore wind power plants, a new reliability indicator for radial grids.

The inter-array grid relates to a significant share of the investments into an offshore wind power plant (OWPP). Optimizing the cable connections regarding costs and reliability is a mathematically complex task due to the high variety of possible wind and component (wind turbine or cable) failure scenarios. This paper presents a novel mixed integer linear programming approach to support investment decisions into OWPPs by trading off cabling purchase and installation costs with power capacity risk (PCR), which is defined as a length-weighed cumulative power flow summation that reflects the consequences of cable failures. Then, quasi-random Monte Carlo simulations assess the optimized collection grids (CGs) to quantify their levelized cost of energy (LCOE). To construct relevant case studies, this work investigates the real OWPPs Ormonde, Horns Rev 1, Thanet, and London Array, which contain 30, 80, 100, and 175 wind turbines. The results reveal Pearson correlation coefficients around 0.99 between the proposed PCR and the expected energy not supplied. Furthermore, this paper's findings indicate that minimum-cost CGs do not necessarily present the lowest LCOE.

A capacitated multi-vehicle covering tour problem on a road network and its application to waste collection

In most Swiss municipalities, a curbside system consisting of heavy trucks stopping at almost each household is used for non-recoverable waste collection. Due to the many stops of the trucks, this strategy causes high fuel consumption, emissions and noise. These effects can be alleviated by reducing the number of stops performed by the collection vehicles. One possibility consists of selecting a subset of candidate locations that are scattered throughout the municipality to place collection points which are used by residents to bring their waste. Provided that the underlying road network is available and that the collection vehicle has a known capacity, we refer to this problem as the capacitated multi-vehicle covering tour problem on a road network (Cm-CTP-R). We propose a road-network-based mixed-integer linear programming (MILP) formulation that exploits the sparsity of the network. We compare it against the MILP formulation that results from assuming a customer-based graph, which is typically used in vehicle routing problems (VRP). To solve large instances, we develop a two-phased heuristic approach that addresses the two subproblems the Cm-CTP-R is built on: a set covering problem to select the locations and a split-delivery VRP to determine the tours. Computational experiments on instances derived from real-life data show that the road-network-based formulation is better suited. Furthermore, the proposed heuristic provides good solutions with optimality gaps below 1.7% and finds better solutions for most of the instances that the exact method is not able to solve within a given time limit.

The Orienteering Problem with Drones

We extend the classical problem setting of the orienteering problem (OP) to incorporate multiple drones that cooperate with a truck to visit a subset of the input nodes. We call this problem the OP with multiple drones (OP-mD). Drones have a limited battery endurance, and thus, they can either move together with the truck at no energy cost for the battery or be launched by the truck onto short flights that must start and end at different customer locations. A drone serves exactly one customer per flight. Moreover, the truck and the drones must wait for each other at the landing locations. A customer prize can be collected at most once, either upon visiting it by the truck or upon serving it by a drone. Similarly to the OP, we maximize the total collected prize under the condition that the truck and the drones return to the depot within a given amount of time. We provide a mixed-integer linear programming formulation for the OP-mD and devise a tailored branch-and-cut algorithm based on a novel decomposition of the problem. We solve instances of the OP-mD with up to 50 nodes within one hour of CPU time with a standard computational setup. Finally, we adapt our framework to solve closely related problems in the literature and compare the resulting computational performance with that of previous studies.