Gurobi Solver Research Articles

A novel cooperative multi-search benders decomposition for solving the hydrothermal unit-commitment problem

Renewable energy and modernization of power operation demand Independent System Operators (ISOs) to solve ever more complex and larger programming problems to securely and economically schedule power resources. A key step in the scheduling process is the unit commitment (UC). In a hydro-dominated system, this process also involves managing reservoirs and is called hydrothermal UC (HTUC). Due to its size, non-convex nature and strict solution-time requirements, HTUC remains a challenging problem to ISOs even in its deterministic form. Relying on a commercial solver to directly tackle this problem might require a careful configuration of the solver’s parameters, and, given the generality of commercial solvers, it might not fully exploit all characteristics of the HTUC. As an alternative, researchers have recently proposed dual-decomposition strategies for addressing the UC, which enables solving subproblems in parallel. However, dualizing a non-convex problem generally results in a duality gap and requires adhoc techniques for recovering a primal solution. In this paper, we propose a novel cooperative multi-search Benders decomposition approach to solve the deterministic HTUC. Our approach benefits from being a primal method and, at the same time, it exploits the structure of the HTUC to solve subproblems in parallel. To assess the effectiveness of our proposal, we use the state-of-the-art solver Gurobi as a benchmark, and we perform our experiments on 25 cases of a large-scale system with over 1,000 generating units and 7,000 buses. On average, our approach is more than 15 times faster than Gurobi and it is able to solve the HTUC problems in under 20 min to a 0.1% gap.

Two-stage stochastic programming approach for limited medical reserves allocation under uncertainties

At the early stage of public health emergencies, when the conventional medical reserves prepared are insufficient, and productivity could temporarily not meet the surge in demand, donations can be used to cover excess demand for medical supplies to a large extent. This paper explicitly considers the allocation problem of limited medical reserves during a public health emergency, incorporating uncertainty in demand and donated supplies and the priorities of health care centers. The problem is formulated as a two-stage stochastic program that regards the donated supplies as an efficient recourse action, aiming to minimize the total losses. The optimal allocation strategy of limited medical reserves and donations is obtained by solving the model using Gurobi solver. Finally, the effectiveness of the proposed approach is verified by a series of computational results, which show that the solutions of our method not only benefit the emergency demand fulfill rate but reduce the total losses as well.

Evaluation of intermittent-distributed-generation hosting capability of a distribution system with integrated energy-storage systems

The penetration rate of distributed generation is gradually increasing in the distribution system concerned. This is creating new problems and challenges in the planning and operation of the system. The intermittency and variability of power outputs from numerous distributed renewable generators could significantly jeopardize the secure operation of the distribution system. Therefore, it is necessary to assess the hosting capability for intermittent distributed generation by a distribution system considering operational constraints. This is the subject of this study. An assessment model considering the uncertainty of generation outputs from distributed generators is presented for this purpose. It involves different types of regulation or control functions using on-load tap-changers (OLTCs), reactive power compensation devices, energy storage systems, and the reactive power support of the distributed generators employed. A robust optimization model is then attained It is solved by Bertsimas robust counterpart through GUROBI solver. Finally, the feasibility and efficiency of the proposed method are demonstrated by a modified IEEE 33-bus distribution system. In addition, the effects of the aforementioned regulation or control functions on the enhancement of the hosting capability for intermittent distributed generation are examined.

Multi-supply multi-capacitated p-median location optimization via a hybrid bi-level intelligent algorithm

The Capacitated P-Median Problem (CPMP) is a classic discrete facility location problem in which p capacitated facilities (medians) are selected from an economic point of view to serve a set of demand vertexes so that the total demand assigned to each of the facilities (medians) does not exceed its capacity. Since the capacity used by each facility is fixed and does not depend on the vertex demand, the traditional capacitated p-median problem’s total capacity is underutilized. To make full use of the total capacity, a more generalized model, namely, the Multi-supply Multi-Capacitated Location Problem (MMCLP), is considered in this paper, which assumes not only that each facility is allowed to open with different capacities while meeting the constraint of total capacity but also that each demand vertex can accept supplies from different facilities. To solve the MMCLP effectively, we propose a hybrid bi-level optimization framework based on decomposition. First, the original MMCLP is decomposed into numerous mixed-integer programming (MIP) sub-problems, i.e., inner optimization is performed, and for outer optimization, a Genetic Algorithm based on Co-Evolution of Population and Neighborhood (GACEPN) is proposed to find better sub-problems near the good sub-problems among all the candidate sub-problems. The inner optimization problem is solved by a commercial MIP solver. The results obtained are returned to the outer layer so that the neighborhood of the outer layer optimization can be adjusted appropriately to generate better sub-problems, thereby approaching the optimal solution of the original problem. The experiments show that for a large-scale MMCLP, compared to the Gurobi solver, the proposed bi-level algorithm (a hybrid of the GACEPN and Gurobi) can find a better solution in less time.

Sustainable closed-loop supply chain network: Mathematical modeling and Lagrangian relaxation

This paper addresses a novel two-stage model for a Sustainable Closed-Loop Supply Chain (SCLSC). This model, as a contribution, provides a balance among economic aims, environmental concerns, and social responsibilities based on price, green quality, and advertising level. Therefore, in the first stage, the optimal values of price are derived by considering the optimal level of advertising and greening. After that, in the second stage, multi-objective Mixed-Integer Linear Programming (MOMILP) is extended to calculate Pareto solutions. The objectives are include maximizing the profit of the whole chain, minimizing the environmental impacts due to CO2 emissions, and maximizing employee safety. Besides, a Lagrangian relaxation algorithm is developed based on the weighted-sum method to solve the MOMILP model. The findings demonstrate that the proposed two-stage model can simultaneously cope with coordination decisions and sustainable objectives. The results show that the optimal price of the recovered product equals 75% of the new product price which considerably encourages customers to buy it. Moreover, to solve the MOMILP model, the proposed algorithm can reach to exact bound with an efficiency gap of 0.17% compared to the optimal solution. Due to the use of this algorithm, the solution time of large-scale instances is reduced and simplified by an average of 49% in comparison with the GUROBI solver.

A stochastic multi-objective optimization framework for distribution feeder reconfiguration in the presence of renewable energy sources and energy storages

In this paper, a multi-objective optimization framework is proposed to solve the distribution feeder reconfiguration (DFR) problem and its operation considering the demand response (DR) program, renewable energy sources (RES's), and electrical energy storages (EES's). The proposed model is implemented on 33-bus and 118-bus radial distribution systems while the uncertainties of RES's output power, load demand and electricity price are taken into account. The Monte Carlo simulation approach is used to generate scenarios while the backward scenario reduction approach is used to reduce the number of scenarios. The studied problem is modeled using the Epsilon-constrained method as a two objective problem and it is solved in the form of five case studies using the GUROBI solver in GAMS software. Our analysis of the results shows that reducing losses and increasing system reliability increases the production of local generation units, thereby increasing the operating costs. In addition, simulation results demonstrate that considering the dynamic topology reduced losses by 9.73% and increased reliability by 4.7%. The results also show that using the DR program reduces LMP by about 20% during peak hour.

Mixed Integer Programming versus Evolutionary Computation for Optimizing a Hard Real-World Staff Assignment Problem

Assigning staff to engagements according to hard constraints while optimizing several objectives is a task encountered by many companies on a regular basis. Simplified versions of such assignment problems are NP-hard. Despite this, a typical approach to solving them consists of formulating them as mixed integer programming (MIP) problems and using a stateof-the-art solver to get solutions that closely approximate the optimum.In this paper, we consider a complex real-world staff assignment problem encountered by the professional service company KPMG, with the goal of finding an algorithm that solves it faster and with a better solution than a commercial MIP solver. We follow the evolutionary algorithm (EA) metaheuristic and design a search heuristic which iteratively improves a solution using domain-specific mutation operators. Furthermore, we use a flow algorithm to optimally solve a subproblem, which tremendously reduces the search space for the EA.For our real-world instance of the assignment problem, given the same total time budget of 100 hours, a parallel EA approach finds a solution that is only 1.7% away from an upper bound for the (unknown) optimum within under five hours, while the MIP solver Gurobi still has a gap of 10.5%.

Benders decomposition for the distributionally robust optimization of pricing and reverse logistics network design in remanufacturing systems

The pricing and reverse logistics network design problem in remanufacturing has attracted considerable attention in recent years due to increasingly serious environmental problems. In this study, we consider a pricing and reverse logistics network design problem with price-dependent return quality uncertainty. To handle the high uncertainty in return quality, we propose a distributionally robust risk-averse model to safeguard the profits of investors in extreme situations. We propose a Benders decomposition approach to solve the proposed model. It is enhanced through valid inequalities, local branching, in-out variant methods and scenario-based aggregated cuts. Computational experiments demonstrate that the distributionally robust model can effectively hedge against high uncertainty and that the enhanced Benders decomposition methods significantly outperform their classical counterparts and the off-the-shelf solver Gurobi. Lastly, managerial insights are explored, and future research directions are outlined.

Finding optimal solutions to several gray pattern instances

In this paper we compare two new binary linear formulations to a standard quadratic binary program for the gray pattern problem and solved all three by the Gurobi solver. One formulation performed significantly better and obtained seven optimal solutions that were not proven optimal before. It is interesting that the formulation that performed best is based on significantly more variables and constraints.

Accelerate the optimization of large-scale manufacturing planning using game theory

This paper studies a real-world manufacturing problem, which is modeled as a bi-objective integer programming problem. The variables and constraints involved are usually numerous and dramatically vary according to the manufacturing data. It is very challenging to directly solve such large-scale problems using heuristic algorithms or commercial solvers. Considering that the decision space of such problems is usually sparse and has a block-like structure, we propose to use decomposition methods to accelerate the optimization process. However, the existing decomposition methods require that the problem has strict block structures, which is not suitable for our problem. To deal with problems with such block-like structures, we propose a game theory based decomposition algorithm. This new method can overcome the large-scale issue and guarantee convergence to some extent, as it can narrow down the search space and accelerate the convergence. Extensive experimental results on real-world industrial manufacturing planning problems show that our method is more effective than the world fastest commercial solver Gurobi. The results also indicate that our method is less sensitive to the problem scale comparing with Gurobi.

Qualifying the benefits of ride-sharing on reducing fleet size

Reducing the number of operating vehicles in cities has enormous significance on mitigating traffic congestion and environment pollution. Ride-sharing is an efficient way to reduce fleet size in urban areas. In this work, we propose two integer programming models to qualify the benefits of ride-sharing on reducing fleet size. The proposed models are solved by commercial solver Gurobi. Then we conduct a series of instances based on trip records of New York City to test the proposed models. Results indicate that without delaying drop-off times, the fleet size when considering ride-sharing remains almost the same as ride-hailing service for high-density travel demand settings. Whereas the fleet size drops sharply as the demand density decreases. In addition, the number of vehicles required is reduced by nearly 30% regardless of order density under ride-sharing assumptions when a slight delay is allowed.

Day-ahead Scheduling of Islanding Microgrids Based on Wind Power Forecast

Wind power, as the main energy source of the island microgrid, has uncertain output. Reasonable setting of TOU price can improve adaptability to changes in wind power output. Based on the time series characteristics of wind speed, the EWMA is used to predict the wind speed day-ahead. Then through the method of distribution superposition fitting and roulette, the day-ahead forecast data of wind power considering power fitting error is obtained. As a reference, a new TOU price division method based on wind power output is proposed. At the same time, P2G and energy storage devices were introduced into the cogeneration system to achieve Gas-electric transformation functions. Combined with the heating and electric demand response of the user side, comprehensively considering the mutual conversion relationship between multiple energies. On the premise of considering economy and environmental protection, the Gurobi solver is called by Yalmip to get the start-stop status of the powerside unit and the user-side unit.

SALSA: Combining branch-and-bound with dynamic programming to smoothen workloads in simple assembly line balancing

We consider a version of the well-known simple assembly line balancing problem (called SALBP-SX) where, given the cycle time and the number of stations, the workloads of the stations are to be leveled according to an adequately defined smoothness index SX. Our index SX involves for each station the quadratic deviation of its workload from the average (or ideal) workload and is therefore closely related to the variance, which is a common measure of dispersion in statistics. Contrary to the existing literature on workload smoothing in ALB, which often treats the optimization of a prespecified smoothness index as a secondary objective, we consider our SX-objective as the single one in order to account for the practical relevance of fair workload distributions and avoiding overloaded bottleneck stations.To optimally solve SALBP-SX, we develop a tailored branch-and-bound procedure. It contains a new station-oriented branching scheme, new lower bound arguments, logical tests and, in particular, a dynamic programming scheme for the pre-calculation of potential workloads, which accelerates the procedure greatly. In comprehensive computational experiments, we show that our method clearly outperforms a class of recently developed task-oriented branch-and-bound procedures and also the mathematical programming solver Gurobi.

A dynamic programming-based local search algorithm for solving the dynamic bipartite drawing problem

The dynamic bipartite drawing problem is a challenging NP-hard combinatorial optimization problem with numerous applications. In this study, we propose a dynamic programming-based local search (DP-LS) algorithm for solving it. Unlike previous metaheuristics reported in the literature, which move just one vertex (or two vertices) at each neighborhood iteration, the proposed DP-LS algorithm selects and performs multiple independent neighborhood moves at the same time to enhance the search efficiency. Generally, starting from a random initial solution, DP-LS iteratively explores the search space by integrating the DP-LS to locate local optima, and then uses a perturbation procedure to escape from the local optima. In addition, the proposed incremental evaluation techniques of the insert and swap moves enhance the efficiency of the neighborhood evaluation. Extensive computational experiments on two sets of 1120 problem instances indicate that the proposed DP-LS is highly competitive with the best-performing algorithms (including the general purpose solver Gurobi) in terms of both solution quality and computational efficiency. We analyzed the dynamic programming mechanism in the proposed DP-LS algorithm to determine its effectiveness (increasing the search efficiency tens of times). Not only can the proposed DP-LS algorithm be used to solve the DBDP, it can also be used as a general methodology for solving other combinatorial optimization problems, especially permutation optimization problems.

Improved whale optimisation algorithm for two-sided disassembly line balancing problems considering part characteristic indexes

An effective method for disassembling large-sized waste products is to utilise two-sided disassembly lines. A mixed-integer programming model for two-sided disassembly lines is constructed in this study. The model considers four optimisation objectives: the number of mated-stations, idle index, demand index, and hazard index. The model redefines the hazard and demand indexes and adds new time constraints to the mated-station. In view of the complexity of the problem, a multi-objective improved whale algorithm is proposed, in which two different crossover operators are designed to simulate the predatory behaviour of whales to improve the efficiency of the solution. Then, a disturbance factor is introduced to reduce the probability of the population falling into a local optimum. The validity of the model and the effectiveness of the algorithm are verified by comparing the calculation results of the GUROBI solver with those of the proposed algorithm for two-sided disassembly line examples of different sizes. Subsequently, the algorithm is used to solve large-scale linear problems, and the results are compared with those of other algorithms to verify the superiority of the proposed algorithm. Finally, the model and algorithm are applied to a two-sided disassembly of an engine, and several optimal allocation schemes are obtained.

How can mixed integer linear programming assist dairy manufacturers by integrating supply decisions and production planning?

PurposeThe dairy industry needs to make important decisions regarding its supply chain. In a context with many available suppliers, deciding which of them will be part of the supply chain and deciding when to buy raw milk is key to the supply chain performance. This study aims to propose a mathematical model to support milk supply decisions. In addition to determining which producers should be chosen as suppliers, the model decides on a milk pickup schedule over a planning horizon. The model addresses production decisions, inventory, setup and the use of by-products generated in the raw milk processing.Design/methodology/approachThe model was formulated using mixed integer linear programming, tested with randomly generated instances of various sizes and solved using the Gurobi Solver. Instances were generated using parameters obtained from a company that manufactures dairy products to test the model in a more realistic scenario.FindingsThe results show that the proposed model can be solved with real-world sized instances in short computational times and yielding high quality results. Hence, companies can adopt this model to reduce transportation, production and inventory costs by supporting decision making throughout their supply chains.Originality/valueThe novelty of the proposed model stems from the ability to integrate milk pickup and production planning of dairy products, thus being more comprehensive than the models currently available in the literature. Additionally, the model also considers by-products, which can be used as inputs for other products.

A Multi-Bus Dispatching Strategy Based on Boarding Control

A multi-bus dispatching strategy is proposed for a ring-shaped road bus transport system, which allows dispatching single bus or multiple buses and incorporates volume dynamics on both buses and stations. Also, the passengers’ perceived waiting time is firstly formulated as one part of the cost function to take passengers’ anxiety into account, and thereby improving the bus quality of service of bus operations. At upstream stations, as many passengers as possible will board the bus, which leads to the less space remaining on the bus and thus the enlongated wait for passengers at downstream stations. With the aim to avoid such phenomenon, the bus boarding control is implemented in the passengers’ boarding process captured by a simultaneous loading model to provide boarding opportunities for the waiting passengers at downstream bus stations. The formulated problem is tackled in two different scenarios, i.e., either with a linear cost or with a nonlinear cost. The linear cost, incorporating the passengers’ actual waiting time and the bus utilization, is firstly converted into a Mixed Integer Linear Programming (MILP) problem, and is solved by the commercial solver Gurobi. With the computational complexity as a concern, two different evolutionary algorithms, Genetic Algorithm (GA) and Harmony Search algorithm (HS), are also adopted to solve the problem in real time. In Scenario 2, the nonlinear cost, integrating the passengers’ perceived waiting time and the bus utilization, is directly solved by both GA and HS. Finally, case studies are provided to illustrate the efficiency of our proposed strategy by comparing with the traditional bus schedule strategies, as well as analyzing the different impacts of the bus loading process when either passengers’ actual waiting time or passengers’ perceived waiting time are taken into account.

A Mixed-Integer Convex Model for the Optimal Placement and Sizing of Distributed Generators in Power Distribution Networks

The optimal placement and sizing of distributed generators is a classical problem in power distribution networks that is usually solved using heuristic algorithms due to its high complexity. This paper proposes a different approach based on a mixed-integer second-order cone programming (MI-SOCP) model that ensures the global optimum of the relaxed optimization model. Second-order cone programming (SOCP) has demonstrated to be an efficient alternative to cope with the non-convexity of the power flow equations in power distribution networks. Of relatively new interest to the power systems community is the extension to MI-SOCP models. The proposed model is an approximation. However, numerical validations in the IEEE 33-bus and IEEE 69-bus test systems for unity and variable power factor confirm that the proposed MI-SOCP finds the best solutions reported in the literature. Being an exact technique, the proposed model allows minimum processing times and zero standard deviation, i.e., the same optimum is guaranteed at each time that the MI-SOCP model is solved (a significant advantage in comparison to metaheuristics). Additionally, load and photovoltaic generation curves for the IEEE 69-node test system are included to demonstrate the applicability of the proposed MI-SOCP to solve the problem of the optimal location and sizing of renewable generators using the multi-period optimal power flow formulation. Therefore, the proposed MI-SOCP also guarantees the global optimum finding, in contrast to local solutions achieved with mixed-integer nonlinear programming solvers available in the GAMS optimization software. All the simulations were carried out via MATLAB software with the CVX package and Gurobi solver.

Robust capacity optimization methods for integrated energy systems considering demand response and thermal comfort

Integrated Energy System can realize the cascade utilization of energy, which improves the utilization of energy efficiently and reduce the carbon emission. Taking into account the uncertainty of multi-energy load and renewable energy forecasting, this paper presents a bi-level robust optimization model with demand response and thermal comfort, for the capacity planning and operation problem of Integrated Energy System. The outer level optimization is planning to find the optimal integrated energy system configuration to minimize economic investment, while the inner level is to robustly optimize the system scheduling to simultaneously reduce carbon emissions and dissatisfaction of residents’ participation in demand response. In the simulation, the Pareto front of the multi-objective problem is obtained via NSGA-II algorithm and Gurobi solver, and the best design plans on the Pareto front selected by Topsis method are analyzed and discussed. The results from three modes are compared in simulation, which illustrates the economic and environmental benefits from demand response and thermal comfort. In addition, the impact of different thermal time scales and forecast uncertainties on integrated energy system planning are also discussed. Finally the sensitivity of the influence of the low carbon grid constrains on the optimization is analyzed.

Optimal scheduling of regional integrated energy system considering integrated demand response

Flexible load can optimize the load curve, which is an important means to promote renewable energy consumption. The peculiarities of electricity, heat, cooling and gas loads are analyzed in this paper, considering the fuzzy degree of human perception of water temperature, the characteristic model of hot water load is established. Considering the fuzzy degree of human perception of ambient temperature, the characteristic model of cooling load is established by using PMV and PPD index. Meanwhile, considering four combinations of cut load, translatable load, transferable load and alternative load, and considering the coupling relationship of composite parts, different response models of load are established respectively. With the minimum cost of the system including operation and compensation costs as the objective function, the optimization scheduling model of regional integrated energy system is established, and the Gurobi solver is used for simulation analysis to solve the optimal output and load response curve of each equipment. The results show that the load curve can be optimized, the flexible regulation ability of the regional integrated energy system can be enhanced, the energy loss of the system can be reduced, and the wind power consumption ability of the system can be increased by considering the integrated demand response.