Mixed Integer Linear Problem 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.

Model for dimensioning borehole heat exchanger applied to mixed-integer-linear-problem (MILP) energy system optimization

This paper introduces three novel approaches to size geothermal energy piles in a MILP, offering fresh perspectives and potential solutions. The research overlooks MILP models that incorporate the sizing of a geothermal borefield. Therefore, this paper presents a new model utilizing a g-function model to regulate the power limits. Geothermal energy is an essential renewable source, particularly for heating and cooling. Complex energy systems, with their diverse sources of heating and cooling and intricate interactions, are crucial for a climate-neutral energy system. This work significantly contributes to the integration of geothermal energy as a vital energy source into the modelling of such complex systems. Borehole heat exchangers help generate heat in low-temperature energy systems. However, optimizing these exchangers using mixed-integer-linear programming (MILP), which only allows for linear equations, is complex. The current research only uses R-C, reservoir, or g-function models for pre-sized borefields. As a result, borehole heat exchangers are often represented by linear factors such as 50 W/m for extraction or injection limits. A breakthrough in the accuracy of borehole heat exchanger sizing has been achieved with the development of a new model, which has been rigorously compared to two simpler models. The geothermal system was configured for three energy systems with varying ground and bore field parameters. The results were then compared with existing geothermal system tools. The new model provides more accurate depth sizing with an error of less than 5 % compared to simpler models with an error higher than 50 %, although it requires more calculation time. The new model can lead to more accurate borefield sizing in MILP applications to optimize energy systems. This new model is especially beneficial for large-scale projects that are highly dependent on borefield size.

On the relationship between the value function and the efficient frontier of a mixed integer linear optimization problem

On the relationship between the value function and the efficient frontier of a mixed integer linear optimization problem

Optimal Scheduling of Source–Load Synergy in Rural Integrated Energy Systems Considering Complementary Biogas–Wind–Solar Utilization

To address the issues of the low usage efficiency and illogical structure in rural regions, this study builds a rural integrated energy system (RIES) that incorporates the complementary use of biogas, wind, and light. For resolving the RIES optimum-low-carbon-economic-dispatch problem, a source–load-cooperative optimal-dispatch strategy is proposed. Firstly, a multi-energy integrated demand response (IDR) model based on time-of-use tariffs and time-varying biogas costs is established on the demand side. Secondly, power-to-gas devices are added on the supply side to optimize the system’s electricity–gas-coupling relationship and increase the wind power output space. Thirdly, an RIES-oriented carbon-trading model is constructed by considering the actual carbon emissions of gas loads and the stepped-carbon-trading mechanism. Finally, an optimal-dispatch model is built with the objective function of reducing the total energy cost, wind abandonment cost, IDR cost, and carbon emission cost, while the problem is transformed into a mixed-integer linear problem and solved using CPLEX 12.9. By setting up four scenarios for example analysis, the results show that on typical days in spring, summer, autumn, and winter, the total operating costs of the stepped-carbon-trading system (Scenario 1), taking into account the source-side power-to-gas (P2G) device and the load-side IDR, are reduced by 12.25%, 11.25%, 12.42%, and 11.56%, respectively, compared to the system without the introduction of the IDR (Scenario 3). In contrast to the system that lacks a P2G device at the source end (Scenario 2), the overall costs are decreased by 4.97%, 3.07%, 5.02%, and 5.36%, but the wind power consumption rates are increased by 11.63%, 7.93%, 11.54%, and 11.65%, respectively. Stepped emission trading (Scenario 1) reduces the total operating costs by 5.12%, 3.15%, 5.21%, and 6.84%, respectively, while reducing the biogas costs by 9.75%, 7.74%, 9.67%, and 9.57%, respectively, in comparison to traditional emission trading (Scenario 4). The example results demonstrate the economics, effectiveness, and reliability of a stepped-carbon-trading system with an integrated P2G load-side energy demand response.

P-facility capacitated location problem with customer-equilibrium decisions: a recreational case study in Mazandaran province

Purpose This paper aims to propose a bi-level mixed integer linear location-allocation problem. The upper-level objective function is dedicated to minimizing the total distances covered by customers to meet the p-selected facilities and the fixed cost values for establishing these facilities. While in the lower level, a customer preference function evaluates the priority of customers in selecting facilities. Design/methodology/approach The solution approach to the proposed model uses the Karush–Kuhn–Tucker (KKT) optimality conditions to the lower-level problem where a set of p-selected facilities are introduced as the selection of the upper-level decision maker. The bi-level model reduces to a single-level model with some added binary variables. Findings Sensitivity analysis of the proposed bi-level model concerning variations of such different parameters as customers’ preferences and the number of selected facilities have been provided, using some numerical examples. Also, locating a recreational facility in Mazandaran province, Iran, has been provided to evaluate the reliability of the proposed model and efficiency of the solution approach, as well. Originality/value To the best of the authors’ knowledge, this paper is original and its findings are not available elsewhere.

A branch-and-cut algorithm for a skip pick-up and delivery problem

In this study we present a branch-and-cut algorithm for a skip pick-up and delivery problem. The study is motivated by a real-life problem in which full skips are transported from waste drop-off stations to treatment facilities where they are emptied, and then brought back to the original drop-off station. The transportation of the skips is done by trucks with the capacity of carrying two containers at a time. The planning problem is to design the routes of the trucks that perform the collection to satisfy a number of requests in a planning period. A truck route starts at the first pick-up, the truck then performs a sequence of pick-ups, treatments, and deliveries, and the route ends at the last delivery. From the truck perspective, the three actions of pick-up, treatment, and delivery can be performed in any order that respects the vehicle capacity of two and the route duration constraint, but for the single request, the three actions must be performed in the stated order. The problem is formulated as a mixed integer linear problem and several classes of valid inequalities are proposed and integrated into a branch-and-cut algorithm.

Fast robust optimization of ORC based on an artificial neural network for waste heat recovery

Uncertainty in the organic Rankine cycle (ORC) system and the highly complex ORC models pose challenges to optimal operation. A data-driven robust parametric optimization for the ORC system is proposed to ensure high and stable performance under uncertainty. The uncertainty of the cyclic variables is estimated by their distribution parameters, and the average (expected) thermodynamic performance (the net output power of the ORC) is maximized as the optimization objective while minimizing its variance. An artificial neural network with the rectified linear unit is used as a surrogate model for optimization. Then the robust parametric optimization problem can be transformed into a mixed-integer linear optimization problem with a chance-constrained form, and the robust optimal solution can be solved quickly. Case studies show that the solution time of the robust optimization problem using the proposed method is about 1.4 s, which is much less than obtaining the optimal solution based on ORC mechanism models. Meanwhile, the optimal operating condition derived by the proposed robust optimization approach outperforms that obtained by the traditional deterministic strategy. The proposed approach not only improves the robustness of the system but also demonstrates the importance of considering uncertainty in parametric optimization.

On the Number of Customer Classes in a Single-Period Inventory System

A common practice in inventory systems with several customers requiring differentiated service levels is to group them into two or three classes, where a customer class is a group of customers with the same preset service level in terms of product availability. However, there is no evidence that grouping customers into two or three classes is optimal in terms of the ordering policy parameters. This paper studies the effect of the number of customer classes on the inventory level of a single-period inventory system with stochastic demand and individual service-level requirements from multiple customer classes. Using a Sample Average Approximation approach, we formulate computationally tractable multi-class service level models, under responsive and anticipative priority policies in cases of shortage, as mixed integer linear problems (MIPs). The effect of the number of classes on the inventory level is determined using a round-up aggregation scheme; i.e., given a sufficiently large initial number of classes, it is reduced by adding the lower service level classes to the next higher class. We analytically characterize the optimal inventory level under responsive and anticipative priority policies as a function of the initial number of classes and the number of classes grouped based on the round-up aggregation scheme. Under a responsive priority policy, we show that there is an optimal number of classes, while under an anticipative priority policy, the optimal number of classes is equal to the initial number of classes. The effect of free-riders resulting from the round-up aggregation scheme on the optimal inventory level is studied through numerical experiments.

Mathematical Programming Formulations for the Berth Allocation Problems in Container Seaport Terminals

Background: Improving the performance of marine terminals is one of the major concerns of both researchers and decision-makers in the maritime transportation sector. The problem of container storage planning and the berth allocation problem (BAP) are the two mainstays of optimizing port operations. Methods: In this work, we address these two issues, proposing two mathematical models that operate sequentially and are applicable to both static and dynamic cases. The first developed model is a mixed-integer linear problem model aimed at minimizing vessel traffic time in the port. The second model developed is a multi-objective optimization model based on goal programming (GP) to minimize both container transfer time and the number of storage areas (minimizing container dispersion). Results: The robustness of the proposed models has been proven through a benchmark with tests using data from the literature and real port data, based on the IBM ILOG CPLEX 12.5 solver. Conclusions: The two developed mathematical models allowed the both minimization of the transfer time and the number of used storage areas, whatever the number of operations handling companies (OHCs) operating in the seaport and for both static and dynamic cases. We propose, as prospects for this work, the development of a heuristic model to deal with the major instances relating to the case of large ports.

Exact and Heuristic Solution Techniques for Mixed-Integer Quantile Minimization Problems

We consider mixed-integer linear quantile minimization problems that yield large-scale problems that are very hard to solve for real-world instances. We motivate the study of this problem class by two important real-world problems: a maintenance planning problem for electricity networks and a quantile-based variant of the classic portfolio optimization problem. For these problems, we develop valid inequalities and present an overlapping alternating direction method. Moreover, we discuss an adaptive scenario clustering method for which we prove that it terminates after a finite number of iterations with a global optimal solution. We study the computational impact of all presented techniques and finally show that their combination leads to an overall method that can solve the maintenance planning problem on large-scale real-world instances provided by the ROADEF/EURO challenge 2020 1 and that they also lead to significant improvements when solving a quantile-version of the classic portfolio optimization problem. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This work was supported by the Deutsche Forschungsgemeinschaft [CRC TRR 154], Fonds De La Recherche Scientifique [PDR T0098.18], and Bundesministerium für Bildung und Forschung. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2022.0105 .

A benchmark of optimization solvers for genome-scale metabolic modeling of organisms and communities.

Genome-scale metabolic modeling is a powerful framework for predicting metabolic phenotypes of any organism with an annotated genome. For two decades, this framework has been used for the rational design of microbial cell factories. In the last decade, the range of applications has exploded, and new frontiers have emerged, including the study of the gut microbiome and its health implications and the role of microbial communities in global ecosystems. However, all the critical steps in this framework, from model construction to simulation, require the use of powerful linear optimization solvers, with the choice often relying on commercial solvers for their well-known computational efficiency. In this work, I benchmark a total of six solvers (two commercial and four open source) and measure their performance to solve linear and mixed-integer linear problems of increasing complexity. Although commercial solvers are still the fastest, at least two open-source solvers show comparable performance. These results show that genome-scale metabolic modeling does not need to be hindered by commercial licensing schemes and can become a truly open science framework for solving urgent societal challenges.IMPORTANCEModeling the metabolism of organisms and communities allows for computational exploration of their metabolic capabilities and testing their response to genetic and environmental perturbations. This holds the potential to address multiple societal issues related to human health and the environment. One of the current limitations is the use of commercial optimization solvers with restrictive licenses for academic and non-academic use. This work compares the performance of several commercial and open-source solvers to solve some of the most complex problems in the field. Benchmarking results show that, although commercial solvers are indeed faster, some of the open-source options can also efficiently tackle the hardest problems, showing great promise for the development of open science applications.

Reliability based computational model for stochastic unit commitment of a bulk power system integrated with volatile wind power

Reliable and economic operation of transmission systems is one of the most onerous problems faced by the grid operators as a penetration rate of wind energy is on rise. It is a crucial carbon-free source and alleviates dependency upon the conventional resources, though its stochastic and unpredictable nature can cause the risk of random component failures. Therefore, reliability evaluation of the system under wind power is of paramount importance and is suggested as the proposed study. A Reliability based Stochastic Unit Commitment (RSUC) model is proposed for a bulk power system integrated with volatile wind power. It is formulated using Mixed Integer Linear problem considering networks and spinning reserve constraints. The volatility of wind speed is modeled by generating different scenarios using K-means clustering approach with computation of correlation among them using autocorrelation factor and their transition probabilities is proposed using Markov Chain (MC). Further, the system's reliability is evaluated using Loss of Load Probability (LOLP) and Expected Energy Not Supplied (EENS). The proposed approach is investigated on the IEEE 24 Reliability Test System (RTS) under the various overloading and random component outage conditions. Further, its results effectiveness is compared and validated with the other existing approaches.

Hierarchical Stochastic Frequency Constrained Micro-Market Model for Isolated Microgrids

With the developments of isolated microgrids (IMGs) and prosumers in remote areas, energy trading has emerged as a critical aspect of IMGs. However, the lack of an upstream network and the low inertia of the system may threaten the secure operation of these networks. This paper proposes a Micro-Market (lM) model for IMGs that includes a precise hierarchical control structure. To address the IMGs low inertia and high intermittency of renewable energy sources (RES), the proposed lM manages the active-reactive power and schedules primary and secondary active reserves to maintain the frequency within in a predefined range. Additionally, a bidirectional linearized AC power flow is established to schedule the reactive reserve and the proposed model is formulated as a two-stage stochastic mixed-integer linear problem (MILP) to maximize social welfare (SW) over the next 24 hours. To validate the effectiveness of the proposed model, the lM is tested on an IMG based on a CIGRE medium-voltage benchmark system, and different operational cases are simulated. The results demonstrate that the proposed model, which takes into account hierarchical control levels and technical issues of the IMG, is a cost-effective way to maximize social welfare while ensuring the secure operation of the IMG.

Lua APIs for mathematical optimization

In this paper we present our Lua programming language libraries that enable modeling and solving mixed integer linear and nonlinear mathematical optimization problems. On one side, these libraries provide a framework for developing algorithms written in Lua that use certain well known solvers within more complex procedures. On the other side, they facilitate the transformation of both input and output data of a mathematical programming problem, in order to compute the standardized coefficients that have to be transmitted to the solvers or prepare solution of solved problems for further data processing. Both of these use cases, together with Lua programming language simplicity, versatility and performance, make Lua a suitable programming language for use in scientific and engineering researches.

A potential framework for cross-zonal reserve procurement on European power exchanges based on robust optimization

In order to prepare for unexpected events in the power system, regulation reserves are routinely procured in advance. Economic considerations dictate that it is advantageous to perform reserve procurement and energy allocation on a joint market platform using a single clearing algorithm. Furthermore, it is also beneficial if the option exists to allocate resources between different bidding zones to alleviate local shortages and corresponding price pressures. The present paper proposes a solution algorithm for this task in a European market environment without unit commitment and uplift payments to non-convex bidders. The main challenge of cross-zonal reserve allocation arises from the fact that deployment decisions are not yet known, therefore the actual deployment flows are uncertain at the time of procurement. The proposed algorithm handles this issue with robust optimization techniques and provides a guarantee of feasible transmission limits for deployment i.e. reserve deliverability. The clearing model is formalized as a computationally efficient Mixed Integer Linear Problem; its exact equations are presented along with case studies for the purpose of demonstration. The research is part of the FARCROSS H2020 project of the European Union.

Math‐based reinforcement learning for the adaptive budgeted influence maximization problem

AbstractIn social networks, the influence maximization problem requires selecting an initial set of nodes to influence so that the spread of influence can reach its maximum under certain diffusion models. Usually, the problem is formulated in a two‐stage un‐budgeted fashion: The decision maker selects a given number of nodes to influence and observes the results. In the adaptive version of the problem, it is possible to select the nodes at each time step of a given time interval. This allows the decision‐maker to exploit the observation of the propagation and to make better decisions. This paper considers the adaptive budgeted influence maximization problem, that is, the adaptive problem in which the decision maker has a finite budget to influence the nodes, and each node requires a cost to be influenced. We present two solution techniques: The first is an approximated value iteration leveraging mixed integer linear problems while the second exploits new concepts from graph neural networks. Extensive numerical experiments demonstrate the effectiveness of the proposed approaches.

Coordination of preventive and emergency dispatch in renewable energy integrated power systems under extreme weather

AbstractIn recent years, frequent occurrence of extreme weather events has caused great losses to the power system. Meanwhile, the high penetration of renewable energy has brought new challenges to the safe operation of the system. Coordination of various measures against different stages of disaster can effectively reduce the overall loss and enhance system's resilience. This paper proposed a methodology to coordinate preventive and emergency dispatch stages based on the defender–attacker–defender model, in which various dispatching measures are modelled as defenders, whereas the extreme weather event and wind power's uncertainty are modelled as attackers. Specifically, a two‐stage three‐layer robust optimal dispatching model is established, in which conventional generator's output, wind farm's grid‐connected planning, and load shedding are taken as the main dispatching measures. To solve this model, the column and constraint generation algorithm is used to transform the three‐layer optimization model into a mixed integer linear problem . The effectiveness of the proposed model is verified in the modified IEEE RTS‐79 test system, and the results show that the coordination of preventive and emergency dispatch can effectively reduce the load outage cost under extreme weather, thus enhancing the ability of power system to cope with extreme scenarios.

Quick taxi route assignment via real-time intersection state prediction with a spatial-temporal graph neural network

The optimization of airport taxi routes is crucial for increasing airport capacity and resource utilization. However, existing methods for preplanning schedules are often time-consuming. To address this, we propose a new approach that reduces the problem scale by predicting the intersection state in the airport taxiway network using a spatial–temporal graph neural network (STGNN) that incorporates a graph convolutional network and recursive neural network with an attention mechanism. A mixed-integer linear problem (MILP) model for taxi route assignment is established based on a subgraph obtained from the prediction results. Our approach treats time as a continuous variable without discretization, allowing aircraft to stop and wait at any node to avoid conflict. Finally, we demonstrate the effectiveness of our approach by testing it on actual operation data from Beijing Capital International Airport (PEK). Compared with several baseline algorithms, the method developed in this study has a significant impact on improving airport transportation efficiency. The results indicate that our method not only produces high-quality solutions but also operates in real time.

Operation of interconnected power and freshwater networks

As freshwater demand continues to rise and conventional freshwater resources decline, the interaction between the freshwater and power sectors is becoming increasingly relevant. Coordinating both systems entail minimizing the operation costs of the power and freshwater system while supplying the demands of electricity and freshwater and complying with operational constraints. This is particularly relevant in Middle Eastern regions, which have limited availability of freshwater resources. Careful coordination of electricity generation and freshwater production is beneficial for both the power and the freshwater systems. For example, producing excess freshwater at hours of low electricity production cost and storing it for usage at hours of high electricity production cost is economically advantageous. We use detailed models for the operation of the power and freshwater systems. That is, the power system operation is represented through a network-constrained unit commitment model, while the freshwater system model includes nonlinear representations of the freshwater production and transportation. The resulting problem is computationally intractable for large-scale systems. To achieve tractability, nonlinearities in the freshwater model are linearized. The final model is a tractable mixed-integer linear optimization problem that is solved using a state-of-the-art branch-and-cut solver. Simulation results demonstrate that a coordinated scheduling results in lower cost and lower peak as compared with uncoordinated scheduling.

Bilevel linear optimization belongs to NP and admits polynomial-size KKT-based reformulations

It is a well-known result that bilevel linear optimization is NP-hard. In many publications, reformulations as mixed-integer linear optimization problems are proposed, which suggests that the decision version of the problem belongs to NP. However, to the best of our knowledge, a rigorous proof of membership in NP has never been published, so we close this gap by reporting a simple but not entirely trivial proof. A related question is whether a large enough “big M” for the classical KKT-based reformulation can be computed efficiently, which we answer in the affirmative. In particular, our big M has polynomial encoding length in the original problem data.