Integer Linear Problem Research Articles

Cleaning scheduling in photovoltaic solar farms with deterministic and stochastic optimization

Soiling in solar panels causes a decrease in their ability to capturing solar irradiance, thus reducing the module’s power output. To reduce losses due to soiling, the panels are cleaned. This cleaning represents a relevant share of the operation and maintenance cost for solar farms, for which there are different types of technologies available with different costs and duration. In this context, this paper proposes a method that allows scheduling the dates on which cleaning generates greater utility in terms of income from energy sales and costs associated with cleaning. For this, two optimization models that deliver a schedule of dates where the best income-cost balance is obtained, are proposed and compared: a deterministic Mixed Integer Linear Problem and a stochastic Markov Decision Process. Numerical results show that both models outperform the baseline case by ∼ 4.6%. A simulator was built and both models were compared to the baseline case for 10,000 rainfall and irradiance scenarios. The stochastic model outperformed both models for all scenarios, thus proving that modeling rainfalls increases profitability in the face of uncertainty.

Discretized Cell Modeling for Optimal Layout of Multiple Tower Cranes

Multiple tower crane layout planning (MTCLP) refers to the selection of the types, quantities, and positions of tower cranes and material storage positions. In a construction project, choosing tower cranes with a larger covering radius and lifting capacity can reduce the transportation time of materials. However, it can also simultaneously raise the rental cost and the risk of crane crashes. Determining an optimal multiple tower crane layout plan to achieve high construction efficiency with low cost is still a challenge. Mathematically, MTCLP is a complex combinatorial problem controlled by a number of variables, such as site configuration, building layout, material storage positions, and workload. To precisely describe irregular shaped sites and buildings, this research partitioned the site into unit cells and proposed a cell-based optimization model. For the optimal crane layout plan, the problem was formulated as a mixed integer linear problem (MILP). The model’s mathematical constraints comprised coverage of the given positions, the crane’s safety distance, sufficient crane capacity, and the utilization ratio. The objectives were to increase the crane coverage ratio while reducing the cost and the working areas overlapped by more than two cranes. Two case studies are presented to demonstrate the effectiveness of the model. The first construction project consisted of seven buildings with irregular shapes. By strictly forbidding the areas to be overlapped by more than three tower cranes, the model successfully obtained the optimal crane layout plan using the Gurobi Optimizer. Compared with the contractor’s layout plan, the optimal plan reduced the rental cost by 10.7% and the area overlapped by more than three tower cranes by 4.56%. In the second project, alternative plans were identified by the Gurobi Optimizer and ε-constraint method to enhance confidence in selecting the ultimate crane layout plan. The optimization procedure involved (1) optimizing the layout plan with minimum crane cost and maximum coverage ratio, (2) identifying all possible layouts as the Pareto fronts when the coverage ratio could not be raised without increasing the crane cost, and (3) minimizing the areas of crane overlap covered by more than three tower cranes. The effectiveness of the proposed model is also proved by comparing the differences in layout plans caused by “regular” and “urgent” lifting demands.

On Carbon Tax Effectiveness in Inducing a Clean Technology Transition: An Evaluation Based on Optimal Strategic Capacity Planning

This paper studies carbon tax effectiveness in inducing a transition to cleaner production when a firm faces different technologies and demands over a planning horizon. To determine carbon tax effectiveness, we propose a model based on strategic capacity production planning under carbon taxes that considers proper performance measures. The model, which is formulated as a mixed integer linear problem (MILP), considers issues that previous works have not studied jointly, and that are relevant in a technological transition, such as machine replacement, workforce planning, and maintenance. The effectiveness measures consider levels of clean production and periods to reach a technological transition. Our computational experiments, based on a real case, have shown that in the absence of carbon taxes, a firm has no incentive to transition to clean technology. Still, the effectiveness of carbon taxes depends on the characteristics of the technology available for the production process and the magnitude of the demand. We include managerial insights aimed at both companies and the environmental authority.

Offering strategy of a price-maker virtual power plant in the day-ahead market

With the rapid increase of renewable energy sources (RESs), the virtual power plant model (VPP) has been developed to integrate RESs, energy storage systems (ESSs), and local customers to overcome the RESs’ disadvantages. When the VPP’s capacity is large enough, it can participate in the electricity market as a price-maker instead of a price-taker to obtain a higher profit. This study proposes a bi-level optimization model to determine the optimal trading strategies of a price-maker VPP in the day-ahead (DA) market. The operation schedule of the components in the VPP is also optimized to achieve the highest profit for the VPP. In the bi-level optimization problem, the upper-level model is maximizing the VPP’s profit while the lower-level model is the DA market-clearing problem. The bi-level optimization problem is formulated as a Mathematical Problem with Equilibrium Constraints (MPEC), reformulated to a Mixed Integer Linear Problem (MILP), then solved by GAMS and CPLEX. This study applies the bi-level optimization model to a test VPP system, including wind plants (WP), solar plants (PV), biogas energy plants (BG), ESSs, and several customers. The maximum power outputs of WP and PV are 100MW and 90MW, respectively. The total installed capacity of BG is 70MW, while the ESS’ rated capacity is 100MWh. The local customers have the highest total consumption of 100MW. In addition to the VPP, four GENCOs and three retailers participate in the DA market. The results show that the market-clearing price varies depending on the participants’ production/consumption quantity and offering/bidding price. However, based on the optimization model, the VPP can take full advantage of WP and PV available power output, choose the right time to operate BG, then obtain the highest profit. The results also show that with the ESS’ rated capacity of 100MWh, the ESS’ rated discharging/charging power increased from 10MW to 50MW will increase VPP’s profit from 45987$ to 49464$. The obtained results show that the proposed model has practical significance

The Chebyshev center as an alternative to the analytic center in the feasibility pump

As a heuristic for obtaining feasible points of mixed integer linear problems, the feasibility pump (FP) generates two sequences of points: one of feasible solutions for the relaxed linear problem; and another of integer points obtained by rounding the linear solutions. In a previous work, the present authors proposed a variant of FP, named analytic center FP, which obtains integer solutions by rounding points in the segment between the linear solution and the analytic center of the polyhedron of the relaxed problem. This work introduces a new FP variant that replaces the analytic center with the Chebyshev center. Two of the benefits of using the Chebyshev center are: (i) it requires the solution of a linear optimization problem (unlike the analytic center, which involves a convex nonlinear optimization problem for its exact solution); and (ii) it is invariant to redundant constraints (unlike the analytic center, which may not be well centered within the polyhedron for problems with highly rank-deficient matrices). The computational results obtained with a set of more than 200 MIPLIB2003 and MIPLIB2010 instances show that the Chebyshev center FP is competitive and can serve as an alternative to other FP variants.

On clustering and interpreting with rules by means of mathematical optimization

In this paper, we make Cluster Analysis more interpretable with a new approach that simultaneously allocates individuals to clusters and gives rule-based explanations to each cluster. The traditional homogeneity metric in clustering, namely the sum of the dissimilarities between individuals in the same cluster, is enriched by considering also, for each cluster and its associated explanation, two explainability criteria, namely, the accuracy of the explanation, i.e., how many individuals within the cluster satisfy its explanation, and the distinctiveness of the explanation, i.e., how many individuals outside the cluster satisfy its explanation. Finding the clusters and the explanations optimizing a joint measure of homogeneity, accuracy, and distinctiveness is formulated as a multi-objective Mixed Integer Linear Optimization problem, from which non-dominated solutions are generated. Our approach is tested on real-world datasets.

Modeling and Solving a Linear Integer Problem (PLNE) for the Optimal Localization of a Hub Air Transport in the WAEMU Zone

In this article, we propose a linear integer model for the optimal location of a hub for air traffic in the WAEMU zone. A hub represents for an airline a base where an essential part of its activities is concentrated. Its location must therefore be judiciously determined. The hub location problem is one of the new and promising areas of research in the field of location theory. In order to satisfy a demand, the location of the hub involves the movement of people, goods between origin destination pairs required. Hubs are applied to reduce the number of transport links between the origin and destination airports The proposed model minimizes the distances and takes into account the flow of passengers registered in the different airports and minimizes the total cost of the transfer via the hub airport. The simulations were made with the programming language Python.

Learning for Spatial Branching: An Algorithm Selection Approach

The use of machine learning techniques to improve the performance of branch-and-bound optimization algorithms is a very active area in the context of mixed integer linear problems, but little has been done for nonlinear optimization. To bridge this gap, we develop a learning framework for spatial branching and show its efficacy in the context of the Reformulation-Linearization Technique for polynomial optimization problems. The proposed learning is performed offline, based on instance-specific features and with no computational overhead when solving new instances. Novel graph-based features are introduced, which turn out to play an important role for the learning. Experiments on different benchmark instances from the literature show that the learning-based branching rule significantly outperforms the standard rules. History: Accepted by Andrea Lodi, Area Editor/Design & Analysis of Algorithms – Discrete. Funding: This work was supported by Ivey Business School (David G. Burgoyne Faculty Fellowship); FEDER [MTM2014-60191-JIN]; Spanish Ministry of Education [FPU Grant 17/02643, FPU Grant 20/01555]; Conselleria de Cultura, Educacion e Universidade [ED431C 2021/24]; Natural Sciences and Engineering Research Council of Canada [Discovery Grant 2017-04185]; Spanish Ministry of Science and Technology [MTM2017-87197-C3] and Spanish Ministry of Science and Innovation [PID2021-124030NB-C32].

Robust scheduling of power system considering social costs and environmental concerns

AbstractOne of the most significant problems of the modern electricity markets is to deal with renewable energy resources (RERs) scheduling. The RER generations face severe stochastic behaviour, such that short‐term scheduling of them is also complicated. To overcome this drawback, using hydro pumped storage units (HPSUs) as a fast response and eco‐friendly technology can help to smooth fluctuations of these types of generations and consequently to appropriately dispatch all generations in the energy and reserve market. This article suggests a stochastic optimisation model to optimally operate thermal power plants as well as HPSUs in the day ahead energy and reserve market. Optimisation aims to minimise operation costs, emissions, and social costs subject to several technical constraints. There is an intrinsic deviation between predicted and actual uncertainty variables in the power system. This article presents a stochastic optimal operation model based on robust optimisation. To improve the flexibility of the proposed market, the curtailed demand as a demand response programme (DRP) is taken into consideration. The CPLEX solver of the GAMS software is used to solve the proposed model which has been formulated as a robust mixed integer linear problem (RMILP). The effectiveness of the proposed model is evaluated by applying the offered model to the 9‐bus test power system.

Low carbon and economic dispatch of the multi-microgrid integrated energy system using CCS-P2G integrated flexible operation method

ABSTRACT The multi-microgrid integrated energy system offers multiple effective approaches to promote clean energy utilization as well as carbon emission reduction, etc. Combined with stepped carbon trading mechanism, it proposes a multi-microgrid (MMG) system with CCS-P2G integrated flexible operation method for optimal scheduling. Firstly, based on the microgrids’ “source-load” characteristic, electricity sharing between the source and load type microgrids can be achieved. Secondly, the CCS-P2G integrated flexible operation method is proposed, i.e. introducing HFC (Hydrogen Fuel Cell) and HES (Hydrogen Energy Storage) in P2G can enhance the utilization effect of hydrogen. Moreover, introducing carbon storage equipment in CCS, which can effectively solve the situation of unsynchronized operation times of CCS and P2G. Finally, introducing stepped carbon trading mechanism can be a further way of limiting carbon emission. Based on the above, MMG’s optimal dispatch model has established with objective of minimizing total operation cost by considering economic performance and carbon emission of the MMG system. Above optimal dispatch model is a mixed integer linear problem, which is solved using CPLEX solver. Simulation results show that electric energy interaction between source-type and load-type microgrids enhances renewable energy utilization rate. Compared to CCS-P2G cooperative operation method, the single-day carbon emissions and total operation cost of multi-microgrid system using CCS-P2G integrated flexible operation method are reduced by 37.02% and 6.80%, respectively. Meanwhile, when the stepped carbon trading mechanism was introduced, multi-microgrid system’s single-day carbon emissions is reduced to 5.40% compared to when normal carbon trading was considered, further constraining MMG’s carbon emissions.

Future Trails for Integer Programming and Relations to Artificial Intelligence

Aims: A review of methods and approaches for solving linear integer problems is presented in this work. These problems are classified as NP-hard optimization algorithms in artificial intelligence.
 Study Design: we have used the Google scholar to collect the data resources from past 5 years to analysis the techniques and methods used in different algorithms in artificial intelligence.
 Methodology: Exact optimum solution for this class of challenges also need use of substantial computer resources. The current direction in which several researcher focuses their efforts to effectively address numerous difficult practical issues is the creation of efficient hybrid techniques that combine in an appropriate way the finest elements of multiple methods (precise or estimated). The approximation algorithms' core heuristic techniques might be classified as constructive algorithms and local-improvement algorithms.
 Results: We examined three artificial intelligence algorithms utilizing the linear integer programming approach. Algorithm based on population It has also been demonstrated that a population of a critical size is necessary for a population-based optimization method to be effective. The genetic algorithm is shown next. The goal value associated with this solution may be utilized to effectively reduce the search tree in bound and branch type integer programming methods. Finally, we analyze the particle swarm optimization (PSO) approach, which demonstrates that In most cases, PSO outperforms the Branch and Bound method in solving such issues quickly. 
 Conclusion: In actuality, integer optimization issues describe a wide spectrum of real-world difficulties. Their population and size are constantly growing. Although while accurate methods for integer issues have substantially improved in recent years, their long runtimes and memory needs make them unsuitable for actual medium and large-scale applications.

Modeling and Solving a Linear Integer Problem (PLNE) for the Optimal Localization of a Hub Air Transport in the WAEMU Zone

In this article, we propose a linear integer model for the optimal location of a hub for air traffic in the WAEMU zone. A hub represents for an airline a base where an essential part of its activities is concentrated. Its location must therefore be judiciously determined. The hub location problem is one of the new and promising areas of research in the field of location theory. In order to satisfy a demand, the location of the hub involves the movement of people, goods between origin destination pairs required. Hubs are applied to reduce the number of transport links between the origin and destination airports. The proposed model minimizes the distances and takes into account the flow of passengers registered in the different airports and minimizes the total cost of the transfer via the hub airport. The simulations were made with the programming language Python.

Integreted strategic and tactical optimization planning of biomass to bioethanol supply chains coupled with operational plan using vehicle routing: A case study in Ethiopia

The optimization of supply chain, a subject of active research in process system engineering, frequently leads to large Mixed Integer Linear or Non-Linear Problems more particularly when applied to realistic industrial cases. In this domain, one main challenge concerns multi scale modeling to accurately integrate the different decision scales (from strategic to operational) and to study the impact of the decision across these scales, in order to effectively contributes to intelligent and coherent decision making. Unfortunately, very few papers cover the three decision levels - strategic (long-term decisions: supply network design), tactical (medium-term decisions: management of network design) and operational (short- term decisions: routing, inventory management…) to have a holistic perspective of the supply chain. Therefore, the first objective of this paper is to model as finely as possible the three decision levels by focusing more particularly on the transport related decision due to their importance on the supply chain performances. As a progress, the Vehicle Routing Problem, one of the most studied problem in logistic and one of the most complex, is integrated into a strategic and tactical model. But, the resulting large-scale MILP model must be solved in a reasonable computational time. To achieve this second objective, we developed a novel iterative multi scale strategy that optimizes the model at two spatial and two temporal scales. This strategy is based on a deep knowledge of the mathematical models and especially of the variables that strongly influence the resolution. The real biofuel supply chain in the Ethiopian country is used to demonstrate the capabilities of the approach, and the improvement in computing time and solution strategy. In addition to the change in the design of the supply chain network, the use of an operational model that finely describes the transport activities shows that the costs related to these activities are often underestimated by a factor between 2 and 3 in the classical models of the literature. The main perspectives of this work are to address the two main shortcomings of the proposal: writing and solving in a reasonable computational time a single model rather than using a decomposition (especially useful for multi-objective optimization) and integrating other decisions at the operational level to not be limited to those related to transportation.

Uniform purchasing price approach for Vietnam wholesale electricity market: modeling and discussing

In Vietnam, the reform of the electricity sector is an urgent need, unbundled and striking to begin a Competitive Electricity Market. The Vietnam electricity day-ahead market model is implemented based on the gross pool model with specific market rules and network constraints. Today, the Vietnam electricity market is operating at the Vietnam Wholesale Electricity Market step but is not yet completed. Due to the long shape of Vietnam territory, an implicit cross-compensation mechanism is applied to create a balance between customers leading to an obstruction to allowing multi-buyers in the electricity market. Therefore, the main aim of this paper is based on the official documents to propose the current mathematical model of the Vietnam electricity day-ahead market using a Mixed Integer Linear Problem, called the Single Buyer Day-Ahead Market model. Then, the Multi Buyer Day-Ahead Market model is implemented to allow multi-buyers to trade in the market. The idea of purchasing uniform prices for multi-buyers using in Italy could be also introduced in the Integrated European Electricity Market in case of large price variations between countries as a mechanism to uniformize the price of energy in Europe for consumers. In this paper, purchasing uniform prices is proposed to provide an explicit cross-compensation mechanism for Vietnam. Eventually, the proposed models are evaluated with four scenarios in two years representing typical days in Vietnam. The results confirm that, on one hand, the proposed market-clearing models can be applied in a real application for any expansion of the market, i.e. the number of suppliers, demands, and bidding areas; on the other hand, the purchasing uniform prices provide an explicit cross-compensation between bidding areas helping to enhance the fairness and the transparency of the market.

Solving discrete linear fractional bilevel programs with multiple objectives at the upper level

<p style='text-indent:20px;'>This paper proposes an exact method to solve an integer linear fractional bilevel problem with multiple objectives at the upper level, designated by <inline-formula><tex-math id="M1">\begin{document}$ IFMOBP $\end{document}</tex-math></inline-formula>. The proposed algorithm generates a set of efficient solutions using a branch and cut algorithm based on a continuous upper level linear fractional problem. Then, the integer optimal solution obtained is tested for optimality of the lower level problem. First, the integer optimal solution of the bilevel problem is sought with a single objective function at each level. After that, an efficient cut is added and new integer solutions are determined. The efficient set is updated each time a candidate bilevel feasible solution non dominated is got and the process ends when there are no unexplored parts of the original domain. The proposed method is based on a dantzig cut to find the next best integer solution of the first objective function of the upper level, an efficient cut to get the set of efficient solutions for the main problem, and the classical branch and bound technique for integer decision variables. After the presentation of the algorithm, a numerical example and computational experiments are provided.</p>

Algorithm for the approximate scheduling of network section's track maintenance in railway traffic control problem

Algorithmic and contributory assistance in resolving railway traffic control problems is being developed. It is founded on presenting practical issues in optimizing declarations using linear programming implements. A novel challenge of determining a pathway control a time interval during which some sectors of the railway network are blocked for reparation work is added to the previously proposed models. Applicable statements, aimed at the synchronized examination for pathway control and a train timetable for a detailed portion of the railway network A mathematical model and an optimized strategy are offered as solutions. The creative situation is simplified to a mixed integer linear problem. To explain potential computing challenges in resolving an issue, a method for obtaining an estimate is planned. It would be founded on creating a rudimentary movement timetable and its consequent revision to the reason for the necessity for railway control. Two algorithms are implemented to get an approximate solution. A basic and adapted train schedule is formed in stages of groupings of trains connected by another origin and target terminals in the first. Steps were performed single train once a time as per the time of preparation for leaving in the second. A mathematical experimentation's findings are provided.

Hybrid particle swarm optimization for pure integer linear solid transportation problem

This paper appends three proposed heuristic algorithms, viz. Initialization Algorithm, Fraction Repair Algorithm and Negative Repair Algorithm, to Particle Swarm Optimization (PSO) and extends its application to Pure Integer Linear Solid Transportation Problem (STP). The chief contribution of the paper is that the proposed Hybrid Particle Swarm Optimization (HPSO) algorithm discretizes the continuous search space of PSO so that it can operate in the discrete solution space of STP with any of the valid parameter settings. The Initialization Algorithm is developed to generate the sufficient number of random solutions for initiating HPSO with a diverse population of given size. Whereas Fraction Repair and Negative Repair Algorithms are developed to ensure that each successive population obtained with HPSO remains in the solution space of STP. The validity of HPSO is examined and illustrated by solving a numerical problem with certain parameter settings. Further, the HPSO demonstrates its ability to obtain an optimal or near optimal solution along with the alternate optimal solution, if it exists. It searches for the solutions even without adhering to the rigid conditions of traditional methods that every solution of STP must have a fixed number of non-zero decision variables. Moreover, the performance of HPSO is tested with three different parameter settings and compared amongst each other to analyze their significance in solving the problem. The performance of HPSO and the variants of parameters are statistically validated through an extensive experimental design.

Coordinated optimization scheduling operation of integrated energy system considering demand response and carbon trading mechanism

The low-carbon economy operation of the integrated energy system can be realized by introducing the demand response and carbon trading mechanism into the optimal scheduling of the integrated energy system. In this paper, an optimal scheduling model based on CCHP and carbon capture device is proposed, which takes into account the demand response of cooling, heating and electricity load and ladder-type carbon trading mechanism. Firstly, a multi-energy and multi-type demand response model based on time-of-use electricity price and incentive mechanism is established, and user satisfaction is used to evaluate it. Then, a carbon trading model of integrated energy system is established considering the actual carbon emissions of the system and the ladder-type carbon trading mechanism. Finally, taking the minimum sum of energy purchase cost, maintenance cost, carbon emission cost and compensation cost as the objective function, combined with the operation constraints of multi-energy flow of integrated energy system, an optimal scheduling model which takes into account both low-carbon and economy is constructed, and the problem is transformed into a mixed integer linear problem and solved by CPLEX. By setting up four scenarios for example analysis, the results show that the system total cost of the ladder-type carbon trading is decreased by 5.9% compared with the traditional carbon trading, and on the basis, the system total operating cost is decreased by 3.1% after considering the user-side DR. The simulation results further show that the introduction of DR and ladder-type carbon trading mechanism can flexibly transfer load, reduce gas purchase and reduce system carbon emissions, which has significant application value.

Group Delay-Aware Scalable Mobile Edge Computing Using Service Replication

The number of individuals and groups of users offloading independent and inter-related computational tasks to mobile edge computing (MEC) servers is rapidly increasing, thus overloading them and raising the risk of service interruptions. Hence, reactive service replication has been suggested to enable individuals and groups of users to access services from remote edge servers, thus guaranteeing system scalability. This paper proposes a task offloading and service replication scheme on local and remote MEC servers. The scheme minimizes the response time of all users while satisfying the delay requirements of user groups in traffic-heavy and multimedia-intense applications (e.g., online gaming, multimedia conferencing, augmenting reality). We formulate an integer linear problem that minimizes the average response time of all users while satisfying the time and time difference constraints of the user groups running the same applications. We then use linear relaxation programming using Lagrangian analysis and solve the problem using a numerical solver. In addition, we compare the optimal solution to distance-based and resource-based greedy approaches. The results demonstrate the merits of our proposed optimized decision scheme compared to these two greedy approaches.

Polynomial Time Reachability Analysis in Discrete State Chemical Reaction Networks Obeying Conservation Laws

In this paper the reachability problem of discrete state Chemical Reaction Networks (d-CRNs) is studied. We consider sub-classes of sub-and superconservative d-CRN network structures and prove that the reachability relation can be decided in polynomial time. We make use of the result that in the studied d-CRN sub-classes, the reachability relation is equivalent to the existence of a non-negative integer solution of the d-CRN state equation. The equivalence implies the reformulation of the reachability problem as integer linear programming decision problem. We show that in the studied classes of d-CRN structures, the state equation has a totally unimodular coefficient matrix. As the reachability relation is equivalent to the non-negative integer solution of the state equation, the resulting integer programming decision program can be relaxed to a simple linear program having polynomial time complexity. Hence, in the studied sub-classes of sub and superconservative reaction network structures, the reachability relation can be decided in polynomial time and the number of continuous decision variables is equal to the number of reactions of the d-CRN.