Mixed Integer Linear Formulation Research Articles

Minimizing delays of patient transports with incomplete information: A modeling approach based on the vehicle routing problem

AbstractWe investigate a challenging task in ambulatory care, the minimizing of delays of patient transports. In practice, a limited number of vehicles is available for non-rescue transports. Furthermore, the dispatcher rarely has access to complete information when establishing a transport plan for dispatching the vehicles. If additional transport is requested on demand then schedules need to be updated, which can lead to long delays. We model the scheduling of patient transports as a vehicle routing problem with general time windows and solve it as a mixed-integer linear problem that is modified whenever additional transport information becomes available. We propose a modeling approach that is designed to determine fair and stable plans. Furthermore, we show that the model can easily be modified when transports need to satisfy additional requirements, e.g., during pandemics, exemplarily the Covid-19 pandemic. To show the applicability and efficiency of our modeling approach, we conduct a numerical study using historical data from the region of Middle Franconia. The results reveal and show that, by applying mathematical optimization—or, to be more precise by solving mixed-integer linear problem formulations—one can significantly decrease delays and have considerable potential for optimized patient transports.

Mixed-Integer Linear Optimization Formulations for Feature Subset Selection in Kernel SVM Classification

Mixed-Integer Linear Optimization Formulations for Feature Subset Selection in Kernel SVM Classification

The Hampered [formula omitted]-Median Problem with Neighbourhoods

This paper deals with facility location problems in a continuous space with neighbours and barriers. Each one of these two elements, neighbours and barriers, makes the problems harder than their standard counterparts. Combining all together results in a new challenging problem that, as far as we know, has not been addressed before, but has applications for inspection and surveillance activities and the delivery industry assuming uniformly distributed demand in some regions. Specifically, we analyse the k-Median problem with neighbours and polygonal barriers in two different situations. None of these problems can be seen as a simple incremental contribution since in both cases the tools required to analyse and solve them go beyond any standard overlapping of techniques used in the separated problems. As a first building block, we deal with the problem assuming that the neighbourhoods are not visible from one another and therefore there are no rectilinear paths that join two of them without crossing barriers. Under this hypothesis, we derive a valid mixed-integer linear formulation. Removing that hypothesis leads to a more general and realistic problem, but at the price of making it more challenging. Adapting the elements of the first formulation, we also develop another valid mixed-integer bilinear formulation. Both formulations rely on tools borrowed from computational geometry that allow to handle polygonal barriers and neighbours that are second-order cone (SOC) representable, which we preprocess and strengthen with valid inequalities. These mathematical programming formulations are also instrumental to derive an adapted matheuristic algorithm that provides good quality solutions for both problems in short computing time. The paper also reports extensive computational experience, counting 2400 experiments, showing that our exact and heuristic approaches are useful: the exact approach can solve optimally instances with up to 50 neighbourhoods and different number of barriers within one hour of CPU time, whereas the matheuristic approach always returns good feasible solutions in less than 300 s.

An optimal expansion planning of power systems considering cycle-based AC optimal power flow

This paper presents a novel mixed-integer linear optimization formulation of the AC network-constrained, cost-based, integrated expansion planning problem. The formulation is used to determine the investment needs per technology including the location and sizing of new generation, energy storage, and transmission network assets in a future low-carbon power system. To reduce the size of the resulting problem, the AC optimal power flow (AC-OPF) model is represented in a compact way using cycle constraints. A bound tightening procedure is also considered to reduce the search space and improve the solver performance by adjusting the voltage bounds within the AC-OPF. Contrary to typically used formulations of the integrated expansion planning problem, the constraints considered here include all main aspects of system operation, namely unit commitment, energy storage system management, AC-OPF, and reactive power compensation. Thus, in this paper, we examine how both the proposed transmission expansion modeling developments and the interrelation of the integrated planning constraints affect the computation of the solution to the expansion planning problem. The performance of this formulation is assessed on the RTS-GMLC test system by computing the expansion plan and comparing it with the results of three other expansion planning formulations most frequently employed in the recent literature to address the integrated expansion planning problem for medium to large-scale systems. Expansion plans are computed and compared for different case studies and multiple scenarios. According to the comparative analysis, neglecting the AC-OPF or the unit commitment constraints can increase the total system costs by 7.10%–9.57% or 6.29%–8.39%, respectively. Unlike other modeling approaches, the proposed approach does not rely on simplifications that impact the quality of the solution. Thanks to the incorporated cycle-based AC-OPF constraints and the consideration of a bound tightening procedure, the computation time is reduced by 17.67%–27.21%.

Integrated location and inventory planning in service parts logistics with last-mile delivery outsourcing

Outsourcing last-mile delivery to express companies provides an innovative strategy for capital goods manufacturers to improve service parts logistics. Given delivery dilemma due to geographically dispersed customers and low-frequency uncertain demands, we develop an integrated location and inventory problem with last-mile delivery outsourcing in a two-echelon service parts logistics. The proposed non-linear mixed integer programming model captures important decisions involving (1) location of local warehouses, (2) customer allocation to opened local warehouses, (3) inventory levels at these local warehouses, (4) outsourcing or self-operating last-mile delivery. We then replace service level constraints that cause nonlinearity with two new sets of constraints, and construct a linear mixed integer formulation that could be solved by GUROBI solver for some small-scale problems. It is still a challenging problem for large scale case, for which we propose an adaptive large neighbourhood search (ALNS) algorithm. Our extensive computational experiments verify the effectiveness of our model and algorithm. Computational experiments’ results indicate that outsourcing last-mile delivery amounting to around 12 % total cost savings on average, compared to self-operating in a real-world large instance from our research partner. Furthermore, we test some key parameters with computational experiments, and provide some managerial implications.

Resource overload problems with tardiness penalty: structural properties and solution approaches

In this paper, we consider a resource overload problem and add a tardiness penalty to the objective function when a prescribed project makespan is exceeded, which enables a trade-off between a balanced resource utilization and a project delay. For the tardiness penalty, we distinguish between a constant and variable delay cost variant. Based on the structural properties of the resource overload problem, we show that the search space of the resource overload problem with tardiness penalty can also be reduced utilizing quasistable schedules. In addition, we discuss the application of these findings to further problems, which include objectives composed of a locally concave and a concave function or a reward structure for an early project completion instead of a tardiness penalty. As solution approaches, we present mixed-integer linear model formulations as well as a novel genetic algorithm with a decoding procedure, which exploits the devised structural properties. The performance of the genetic algorithm is improved by implementing learning methods and utilizing lower bounds. Finally, we present results from experiments on small to medium sized problem instances.

Linear or mixed integer programming in long-term energy systems modeling – A comparative analysis for a local expanding heating system

Most computer models used in energy systems optimization modeling studies are formulated using linear equations. However, since linear formulations do not always well reflect real-world conditions, they may not always be adequate as policy and support tools. This is particularly the case for local system studies attempting to represent technologies at the individual scale, as in the case for local heating system modeling. Thus, the aim of this paper is to investigate differences in the resulting heating solutions and model solution times for a local expanding heating system. Three different investment cost structures for individual and district heating solutions for the heating of new housing are investigated using linear and mixed integer linear programming. The results show that the use of district heating is higher for the cost structures that use mixed integer linear programming than it is for the linear cost structures. This result is attributed mainly to the fact that individual air-to-water heat pumps benefit from the linear equation formulation due to its high coefficient of performance during summertime. This finding is important to consider when modeling local energy systems. The solution time is, however, significantly shorter for the linear formulations than for the mixed integer linear formulations.

On relaxations of the max k-cut problem formulations

A tight continuous relaxation is a crucial factor in solving mixed integer formulations of many NP-hard combinatorial optimization problems. The (weighted) max k-cut problem is a fundamental combinatorial optimization problem with multiple notorious mixed integer optimization formulations. In this paper, we explore four existing mixed integer optimization formulations of the max k-cut problem. Specifically, we show that the continuous relaxation of a binary quadratic optimization formulation of the problem is: (i) stronger than the continuous relaxation of two mixed integer linear optimization formulations and (ii) at least as strong as the continuous relaxation of a mixed integer semidefinite optimization formulation. We also conduct a set of experiments on multiple sets of instances of the max k-cut problem using state-of-the-art solvers that empirically confirm the theoretical results in item (i). Furthermore, these numerical results illustrate the advances in the efficiency of global non-convex quadratic optimization solvers and more general mixed integer nonlinear optimization solvers. As a result, these solvers provide a promising option to solve combinatorial optimization problems. Our codes and data are available on GitHub.

Portfolio optimization through a network approach: Network assortative mixing and portfolio diversification

The paper deals with the classical problem of selecting a portfolio in the financial market and follows a risk-return optimization approach. The main issue in portfolio selection is capturing the dependency structure of the returns of the different assets. In the well-known Markowitz models this is measured by the variance/covariance matrix of the assets’ returns. Recent works have focused on a new way of modeling the dependency between returns of different assets by means of the so called “market graph” or “correlation graph”.Basing on this graph, the paper introduces new mixed integer linear formulations for the portfolio selection problem in which the portfolio expected return is maximized, while controlling the worst case loss. There are two innovative aspects related to the market graph. One is the introduction of a set of constraints over the neighborhood of each node. These constraints force the model to include in the portfolio a subset of assets which correspond to nodes poorly connected in the correlation graph. The second is the use of a measure from the area of complex network analysis called ‘assortativity’. A local assortativity coefficient is computed for each node, and it is used to formulate linear objective functions in the optimization models. We propose two Mixed Integer Linear Programs for finding the most ‘disassortative’ portfolios by using either the ‘local degree’ or the ‘local strength’ assortativity coefficients.A set of computational experiments is performed over real-world data sets to test the performance of the proposed models. The experimental results show that combining the assortativity criterion and the neighborhood constraints leads to an effective strategy to obtain portfolios with a good out-of-sample performance under the risk-return viewpoint.

Risk-assessment of carbon-dioxide recycling in a gas-fired power plant using CVaR-based convex optimization

With the rise of renewable resources’ penetration, the use of clean excessive renewable energy as the primary source of a Power to Gas (PtG) system for the reduction of CO2 pollution is quite advantageous. This paper presents a mixed integer linear formulation for modeling the optimal operation of a gas-fired power plant (GFPP)-PtG system that utilizes an excessive renewable resource, wind power, as the primary mover of the proposed PtG system for capturing and recycling the produced CO2 pollution. CO2 and CH4 gas storages are considered in the proposed PtG system to boost the operational efficiency of the system. Additionally, the scenario generation method models both wind power generation and electrical demand. The effective conditional value at risk (CVaR) method, which is a convex risk modeling approach, models the risks imposed by the generated scenarios for wind power and electrical demand. Due to the convexity of the proposed formulation for the PtG system and the risk modeling method, the ultimate model can be simply solved by using any available LP solver. To evaluate the outcomes of an optimization problem based on CVaR, a comparative analysis is conducted with chance constraints programming, which demonstrates the optimality and computational efficiency of the CVaR results due to convexity. The effectiveness of the proposed model is approved by modeling a test system based on real data, and the obtained numerical results are discussed comprehensively.

On the convex hull of convex quadratic optimization problems with indicators

We consider the convex quadratic optimization problem in Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^{n}$$\\end{document} with indicator variables and arbitrary constraints on the indicators. We show that a convex hull description of the associated mixed-integer set in an extended space with a quadratic number of additional variables consists of an (n+1)×(n+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(n+1) \ imes (n+1)$$\\end{document} positive semidefinite constraint (explicitly stated) and linear constraints. In particular, convexification of this class of problems reduces to describing a polyhedral set in an extended formulation. While the vertex representation of this polyhedral set is exponential and an explicit linear inequality description may not be readily available in general, we derive a compact mixed-integer linear formulation whose solutions coincide with the vertices of the polyhedral set. We also give descriptions in the original space of variables: we provide a description based on an infinite number of conic-quadratic inequalities, which are “finitely generated.” In particular, it is possible to characterize whether a given inequality is necessary to describe the convex hull. The new theory presented here unifies several previously established results, and paves the way toward utilizing polyhedral methods to analyze the convex hull of mixed-integer nonlinear sets.

A Mixed-Integer Linear Formulation for a Dynamic Modified Stochastic p-Median Problem in a Competitive Supply Chain Network Design

Background: The Dynamic Modified Stochastic p-Median Problem (DMS-p-MP) is an important problem in supply chain network design, as it deals with the optimal location of facilities and the allocation of demand in a dynamic and uncertain environment. Methods: In this research paper, we propose a mixed-integer linear formulation for the DMS-p-MP, which captures the key features of the problem and allows for efficient solution methods. The DMS-p-MP adds two key features to the classical problem: (1) it considers the dynamic nature of the problem, where the demand is uncertain and changes over time, and (2) it allows for the modification of the facility locations over time, subject to a fixed number of modifications. The proposed model uses robust optimization in order to address the uncertainty of demand by allowing for the optimization of solutions that are not overly sensitive to small changes in the data or parameters. To manage the computational challenges presented by large-scale DMS-p-MP networks, a Lagrangian relaxation (LR) algorithm is employed. Results: Our computational study in a real-life case study demonstrates the effectiveness of the proposed formulation in solving the DMS p-Median Problem. The results show that the number of opened and closed buildings remains unchanged as the time horizon increases due to the periodic nature of our demand. Conclusions: This formulation can be applied to real-world problems, providing decision-makers with an effective tool to optimize their supply chain network design in a dynamic and uncertain environment.

Demand response for flat nonlinear MIMO processes using dynamic ramping constraints

Volatile electricity prices make demand response attractive for processes that can modulate their production rate. However, if nonlinear dynamic processes must be scheduled simultaneously with their local multi-energy system, the resulting scheduling optimization problems often cannot be solved in real time. For single-input single-output processes, the problem can be simplified without sacrificing feasibility by dynamic ramping constraints that define a derivative of the production rate as the ramping degree of freedom. In this work, we extend dynamic ramping constraints to flat multi-input multi-output processes by a coordinate transformation that gives the true nonlinear ramping limits. Approximating these ramping limits by piecewise affine functions gives a mixed-integer linear formulation that guarantees feasible operation. As a case study, dynamic ramping constraints are derived for a heated reactor-separator process that is subsequently scheduled simultaneously with its multi-energy system. The dynamic ramping formulation bridges the gap between rigorous process models and simplified process representations for real-time scheduling.

Exact and metaheuristic approaches to solve the integrated production scheduling, berth allocation and storage yard allocation problem

This work studies an industrial complex specialized in producing and exporting fertilizers. The integration of the production scheduling, berth allocation, and storage yard allocation decisions is necessary to improve the overall performance of the complex as these problems are interrelated. Exact and metaheuristic approaches are proposed to solve the problem. First, a mixed-integer linear formulation along with a branch-and-cut procedure are proposed. The numerical experiments show that these exact methods solve small-sized instances optimally but fail to provide good quality solutions for industrial-sized instances. Therefore, a multi-start GRASP-ILS is proposed where initial solutions are constructed using a greedy randomized adaptive search procedure (GRASP) and improved using an iterated local search algorithm (ILS). The metaheuristic approach is based on a disjunctive graph representation and additional conditions to respect the storage yard constraints and other operational constraints. The results show that the multi-start GRASP-ILS can be effectively used to solve larger instances of the problem.

Team orienteering problem with nonidentical agents and balanced score

In this work, we study a variant of the team orienteering problem motivated by a real-world situation faced by a Mexican telecommunications company. The problem consists of the daily assignment and scheduling of service orders to crews, aiming to balance their wages, and considering the compatibility between service orders and crews. We present a mixed-integer linear formulation with two different metrics to achieve balanced scores and two valid inequalities leveraging the structure of the problem. Afterward, we propose a practical adaptive multi-start heuristic that integrates the learning mechanism of a reactive Greedy Randomized Adaptive Search Procedure. We test the performance of the models and the proposed algorithm on a benchmark of instances adapted from the literature and in a case study based on real data. The results confirm the effectiveness of the proposed algorithm to support the decision-making process.

A parameterized lower bounding method for the open capacitated arc routing problem

Consider an undirected graph with demands scattered over the edges and a homogeneous fleet of vehicles to service the demands. In the open capacitated arc routing problem (OCARP) the objective is to find a set of routes that collectively service all demands with the minimum cost. Each vehicle has limited capacity and it can start and finish the route at any node. The OCARP is NP-hard, and its applications include meter reading and cutting path determination problems. State-of-the-art solution methods developed for the OCARP are heuristics, which show good tradeoffs between solution quality and processing time, but do not provide optimality certificates of the obtained solutions. This work focuses on a lower bounding method for the OCARP which can be used to better assess the quality of heuristic solutions. We propose the Relaxed Flow method (RF(k)) which involves the resolution of a mixed integer linear formulation where all vehicles' capacities are modeled as flows on an augmented graph. A parameter k controls the model tightness and RF(k) is shown to be at least as tight as the well-known Belenguer and Benavent's formulation for any k⩾0. To strengthen the model, capacity cuts were included in RF(k) by means of a branch-and-cut framework. Extensive computational experiments conducted on a set of benchmark instances revealed that our method outperformed previous methods. Computational experiments also demonstrated the importance of the parameterization technique to obtain good results. The previously known lower bounds were improved substantially and optimality certificates were attained in 78.9% of the instances. As far as we know this is the first parameterized lower bounding method proposed for an arc routing problem, and we argue it can be generalized to other variants of arc routing problems and general routing problems.

A compact quadratic model and linearizations for the minimum linear arrangement problem

Given an undirected connected graph G = ( V , E ) , the minimum linear arrangement problem (MinLA) consists in determining a permutation π ≔ ( π 1 , … , π n ) of the node-set V = { 1 , … , n } , which minimizes the sum of edge costs | π i − π j | over all edges { i , j } ∈ E . We introduce a compact quadratic formulation, prove its correctness, and generate new mixed-integer linear formulations that require a very small number of variables and constraints. The idea behind the way we model permutations allows the development of strong valid constraints to strengthen the new formulations. We also adapt some cutting plane procedures from the literature to one of the new models. Computational experiments with benchmark and new instances are very encouraging and improve state-of-the-art solution approaches from the literature. • We investigate the minimum linear arrangement problem. • We propose three new compact formulations for the problem. • We develop strong constraints to strengthen the new formulations. • Theoretical results can be extended to handle problems with a permutational structure.

A new approach for production project scheduling with time-cost-quality trade-off considering multi-mode resource-constraints under interval uncertainty

ABSTRACT Due to today’s competitive environment and information boom, companies are concerned about production planning in uncertain conditions. This paper integrates decision-making method with production scheduling model by considering limited resources. In this paper, a new mathematical model is extended for production project scheduling with multiple execution modes. The main aim of the formulation is to plan and schedule real production projects in uncertain environments. A new mixed-integer linear formulation is presented by considering trade-off of cost, time, as well as quality. Cost–time-quality trade-off is extended with the interval information. In the presented model, activity quality could be enhanced by reworking. The interval forms of some parameters, including duration, quality of activities, cost, and total available resources, are obtained by determining weights of experts and aggregating them. The presented group decision-making method is based on a bi-directional projection measure to deal with interval information. Since the mathematical model is multi-objective and some data are interval, a new modified solution method is developed for solving the model. The presented method for both decision-making and mathematical models is investigated by a real-world production project and two datasets to ascertain the accuracy of the model. Finally, an appropriate sensitivity analysis is proposed.

A Hybrid Evolutionary Algorithm for the Diversified Top-k Weight Clique Search Problem (Student Abstract)

The diversified top-k weight clique (DTKWC) search problem is an important generalization of the diversified top-k clique search problem, which extends the DTKC search problem by taking into account the weight of vertices. This problem involves finding at most k maximal weighted cliques that cover maximum weight of vertices with low overlapping in a given graph. In this study, a mixed integer linear program constraint formulation is proposed to model DTKWC search problem and an efficient hybrid evolutionary algorithm (HEA-D) based on some heuristic strategies is proposed to tackle it. Experiments on two sets of 110 graphs show that HEA-D outperforms the state-of-art methods.

Energy-Aware Model Predictive Control of Assembly Lines

This paper presents a model predictive approach to the energy-aware control of tasks’ execution in an assembly line. The proposed algorithm takes into account both the need for optimizing the assembly line operations (in terms of the minimization of the total cycle time) and that of optimizing the energy consumption deriving from the operations, by exploiting the flexibility added by the presence of a local source of renewable energy (a common scenario of industries that are often equipped, e.g., with photovoltaic plants) and, possibly, also exploiting an energy storage plant. The energy-related objectives we take into account refer to the minimization of the energy bill and the minimization of the peaks in the power injected and absorbed from the grid (which is desirable also from the perspective of the network operator). We propose a mixed-integer linear formulation of the optimization problem, through the use of H-infinite norms, instead of the quadratic ones. Simulation results show the effectiveness of the proposed algorithm in finding a trade-off that allows keeping at a minimum the cycle time, while saving on the energy bill and reducing peak powers.