Mixed-integer Programming Method Research Articles

A fully polynomial time approximation scheme for a stop schedule of a transportation line without branches

For a transportation line without branches, passenger service scheduling is mainly concerned with the determination of stop schedules for planned service trips. A mixed-integer programming method is proposed to maximize customer satisfaction by minimizing the total number of in-vehicle passengers during trips. A heuristic algorithm is proposed for a given number of service trips, in which the error bound of the heuristic is restricted by the number of stations. When the given number of service trips is sufficient to provide express services for all origin–destination pairs, the heuristic provides an optimal solution. However, the error bound of the heuristic approaches infinity as the number of stations increases. A fully polynomial time-approximation algorithm is proposed, where the number of service trips is a decision variable. Finally, the proposed algorithm was applied to the stop-schedule problem of a river bus operating along the Han River in Seoul, Korea.

Peer-to-peer energy trading optimization for community prosumers considering carbon cap-and-trade

With the increasing penetration of distributed renewable energy, prosumers, a kind of customers capable of producing power, are able to share the information of the power consumption and the power price in a community, which evolves into an energy trading between producers and consumers over time. In this paper, considering the benefit conflict whether paying for the power from green power market, or suffering the cost from electricity market together with carbon market due to the carbon cap, a game-theoretic approach is proposed for the prosumers, who have to face the carbon emission constraints, in order to optimize their own utility with an energy trading in a community. It is considered that producers provide the optimal power price for the consumers based on the non-cooperative game, then the consumers decide which producer to purchase power from according to the evolutionary game, afterwards the interaction between the producers and the consumers are analysed in terms of the Stackelberg game. The low-carbon energy trading process in a community is solved with a mixed integer programming method based on the equilibrium of the games. Simulations show that the proposed approach is able to significantly improve the utility of the prosumers in the community.

Buffer Times Between Scheduled Events in Resource Assignment Problem: A Conflict-Robust Perspective

Problem definition: In many resource scheduling problems for services with scheduled starting and completion times (e.g., airport gate assignment), a common approach is to maintain appropriate buffer between successive services assigned to a common resource. With a large buffer, the chances of a “crossing” (i.e., a flight arriving later than the succeeding one at the gate) will be significantly reduced. This approach is often preferred over more sophisticated stochastic mixed-integer programming methods that track the arrival of all the flights to infer the number of “conflicts” (i.e., a flight arriving at a time when the assigned gate becomes unavailable). We provide a theoretical explanation, from the perspective of robust optimization for the good performance of the buffering approach in minimizing not only the number of crossings but also the number of conflicts in the operations. Methodology/results: We show that the buffering method inherently minimizes the worst-case number of “conflicts” under both robust and distributionally robust optimization models using down-monotone uncertainty sets. Interestingly, under down-monotone properties, the worst-case number of crossings is identical to the worst-case number of conflicts. Using this equivalence, we demonstrate how feature information from flight and historical delay information can be used to enhance the effectiveness of the buffering method. Managerial implications: The paper provides the first theoretical justification on the use of buffering method to control for the number of conflicts in resource assignment problem. Funding: This work was supported by the 2019 Academic Research Fund Tier 3 of the Ministry of Education-Singapore [Grant MOE-2019-T3-1-010] and the Research Grants Council of Hong Kong SAR, China [Grant PolyU 152240/17E]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2022.0572 .

Reinforcement learning and mixed-integer programming for power plant scheduling in low carbon systems: Comparison and hybridisation

Decarbonisation is driving dramatic growth in renewable power generation. This increases uncertainty in the load to be served by power plants and makes their efficient scheduling, known as the unit commitment (UC) problem, more difficult. UC is solved in practice by mixed-integer programming (MIP) methods; however, there is growing interest in emerging data-driven methods including reinforcement learning (RL). In this paper, we extensively test two MIP (deterministic and stochastic) and two RL (model-free and with lookahead) scheduling methods over a large set of test days and problem sizes, for the first time comparing the state-of-the-art of these two approaches on a level playing field. We find that deterministic and stochastic MIP consistently produce lower-cost UC schedules than RL, exhibiting better reliability and scalability with problem size. Average operating costs of RL are more than 2 times larger than stochastic MIP for a 50-generator test case, while the cost is 13 times larger in the worst instance. However, the key strength of RL is the ability to produce solutions practically instantly, irrespective of problem size. We leverage this advantage to produce various initial solutions for warm starting concurrent stochastic MIP solves. By producing several near-optimal solutions simultaneously and then evaluating them using Monte Carlo methods, the differences between the true cost function and the discrete approximation required to formulate the MIP are exploited. The resulting hybrid technique outperforms both the RL and MIP methods individually, reducing total operating costs by 0.3% on average.

A sequential mixed-integer programming method for concurrent optimization of core topology and face sheet thickness of a sandwich beam

A method is proposed that allows for the concurrent optimization of core topology and face sheet thickness of a sandwich beam under compliance constraints. The problem is solved using a novel mixed-linear extension of the Topology Optimization of Binary Structure (TOBS) topology optimization method aiming to minimize the total mass of the beam. The method has been demonstrated on a clamped beam example and the results have been compared to results from topology optimization of the core with a range of a priori fixed face sheet thicknesses. It is shown that the new method, starting from a fully populated core, finds a minimum mass that is lower than but in the neighbourhood of the best results from the topology optimization with fixed face sheet thicknesses. By varying the compliance constraint it is shown that the core topology approaches an ideal corrugated geometry as the compliance constraint is relaxed. The trends observed in the results are compared to analytical models for an idealized core.

An enlarged polygon method without binary variables for obstacle avoidance trajectory optimization

In this article, an Enlarged Polygon/Polyhedron (ELP) method without binary variables is proposed to represent the Convex Polygonal/Polyhedral Obstacle Avoidance (CPOA) constraints in trajectory optimization. First, the equivalent condition of a point outside the convex set is given and proved rigorously. Then, the ELP condition describing the CPOA constraints equivalently is given without introducing binary variables, and its geometric meaning is explained. Finally, the ELP method is used to transform the CPOA trajectory optimization problem into an optimal control problem without binary variables. The effectiveness and validity of ELP method are demonstrated through simulations with both simple linear dynamic model (unmanned aerial vehicle) and complex nonlinear dynamic model (hypersonic glide vehicle). Comparison indicates the computational time of ELP method is only 1%-20% of that of the traditional Mixed-Integer Programming (MIP) method.

A successive linearization based optimal power-gas flow method

Optimal power-gas flow (OPGF) problem is a fundamental coordinated operation problem. However, the nonlinear AC power flow model and nonconvex gas flow model pose a huge challenge to OPGF calculation. The common treatments in the existing literature adopt simplified modeling of power and gas systems to obtain a relatively tractable model, which may produce an inaccurate and insecure solution (e.g., violation of voltage limits). Therefore, this paper considers the nonlinear AC power flow modeling and nonconvex gas flow modeling in the OPGF problem. To address the nonconvexities arising from the nonlinear AC power flow model, nonconvex Weymouth equations, and compressors’ gas consumption equations, the concept of successive linearization is introduced and then a successive mixed-integer quadratic programming (SMIQP) method is developed to solve the nonconvex OPGF problem. Numerical results demonstrate the effectiveness of the proposed SMIQP-based OPGF method in terms of high accuracy and high computational efficiency.

Cooperative multi-site EMS sharing EV batteries based on model predictive control

This study investigates a community energy management system (CEMS) composed of multiple individual EMSs, which minimize the cost of electricity for an entire community by sharing the onboard storage batteries of electric vehicles (EVs) among the multiple EMSs. Because EVs move between sites in the community, the EMSs need to cooperate in recharging and discharging of EV batteries with considering the movement of EVs. In this study, we propose a model predictive control of charging and discharging for EV batteries shared by multiple EMSs based on a mixed-integer quadratic programming method and a quadratic programming method. In addition, the electricity price of trade between an EV and an EMS is derived using the Lagrange multiplier in the optimization problem. Finally, the validity of the proposed system is verified using simulations.

Cost-Aware DU Placement and Flow Routing in 5G Packet xHaul Networks

Packet-switched xHaul networks based on Ethernet technology are considered a promising solution for assuring convergent, cost-effective transport of diverse radio data traffic flows in dense 5G radio access networks (RANs). A challenging optimization problem in such networks is the placement of distributed processing units (DUs), which realize a subset of virtualized baseband processing functions on general-purpose processors at selected processing pool (PP) facilities. The DU placement involves the problem of routing of related fronthaul and midhaul data flows between network nodes. In this work, we focus on developing optimization methods for joint placement of DUs and routing of flows with the goal to minimize the overall cost of PPs activation and processing in the network, which we refer to as the PPC-DUP-FR problem. We account for limited processing and transmission resources as well as for stringent latency requirements of data flows in 5G RAN. The latency constraint makes the problem particularly difficult in a packet-switched xHaul network since it involves the non-linear and dynamic estimation of the latencies caused by buffering of packets in the switches. The latency model that we apply in this work is based on worst-case calculations with improved latency estimations that skip from processing the co-routed, but non-affecting flows. We use a mixed-integer programming (MIP) approach to formulate and solve the PPC-DUP-FR optimization problem. Moreover, we develop a heuristic method that provides optimized solutions to larger PPC-DUP-FR problem instances, which are too complex for the MIP method. Numerical experiments performed in different network scenarios indicate on the effectiveness of the heuristic in solving the PPC-DUP-FR problem. In particular, the heuristic achieves up to 63% better results than MIP (at the MIP optimality gap equal to 76%) in a medium-size mesh network, in which the MIP problem is unsolvable for higher traffic demands within reasonable runtime limits. In larger networks, MIP is able to provide some results only for the PPC-DUP-FR problem instances with very low traffic demands, whereas the solutions generated by the heuristic are at least 83% better than the ones achieved with MIP. Also, the analysis performed shows a significant impact of the PP cost factors considered and of the level of cost differentiation of PP nodes on the overall PP cost in the network. Finally, simulation results of a case-study packet xHaul network confirm the correctness of the latency model used.

Applications of Metaheuristics Inspired by Nature in a Specific Optimisation Problem of a Postal Distribution Sector

This paper presents a logistics problem, related to the transport of goods, which can be applied in practice, for example, in postal or courier services. Two mathematical models are presented as problems occurring in a logistics network. The main objective of the optimisation problem presented is to minimise capital resources (Capex), such as cars or containers. Three methods are proposed to solve this problem. The first is a method based on mixed integer programming (MIP) and available through the CPLEX solver. This method is the reference method for us because, as an exact method, it is guaranteed to find the optimal solution as long as the problem is not too large. However, the logistic problem under consideration belongs to the class of NP-complete problems and therefore, for larger problems, i.e., for networks of large size, the MIP method does not find any integer solution in a reasonable computational time. Therefore, two nature-inspired heuristic methods have been proposed. The first is the evolutionary algorithm and the second is the artificial bee colony algorithm. Results indicate that the heuristics methods solve instances of large size, giving suboptimal solutions and therefore, enabling their application to real-life scenarios.

Near-Optimal Covering Solution for USV Coastal Monitoring using PAES

This paper addresses a multi-objective optimization problem for marine monitoring using USV. The objectives are to cover the maximum area with the lowest energy cost while avoiding collisions. The problem is solved using an exact and heuristic methods. First, a multi-objective Mixed Integer Programming formulation is proposed to model the USV monitoring problem. It consists of a combination of the Covering Salesman Problem (CSP) and Travelling Salesman Problem with Profit (TSPP). Then, we use CPLEX software to provide exact solutions. On the other hand, a customized chromosome-size algorithm is used to find heuristic solution. The latter is a multi-objective evolutionary algorithm known as Pareto Archived Evolution Strategy (PAES). The obtained results showed that the exact solving of the USV monitoring mission problem with mixed-integer programming (MIP) methods needs extensive computational costs. However, the customized PAES was able to provide Near-optimal solutions for large-size graphs in much faster time as compared to the exact one.

A bi-objective approach based on fuzzy logic for multi-period technician routing and scheduling problems

This paper deals with a multi-period technician routing and scheduling problem. The problem carries out the tasks at various locations by being divided into teams considering differently skilled technicians. This paper provides a bi-objective mixed integer programming method to model the problem since the goals of the performed model are to optimize the travel cost while simultaneously minimizing the overtime and the waiting time. The paper introduces a fuzzy logic approach to solving the problem by offering a single Pareto solution to satisfy both objectives, rather than providing efficient solutions as in the case of classical multi-objective models. Finally, the paper presents the computational experiments and analyses to evaluate the efficiency of the suggested mathematical formulation and solution approach. The results demonstrate that the performed model accomplishes an acceptable satisfaction level.

OSNR-Aware Modeling and Optimization of Multi-Core Fiber-Based Spectrally–Spatially Flexible Optical Networks

We focus on the optical signal-to-noise ratio (OSNR)-aware optical path (lightpath) planning problem in spectrally and spatially flexible optical networks (SS-FONs) connected using weakly coupled multi-core fibers (MCFs) in which distance-adaptive and super-channel transmission is realized using multiple modulation formats. In the SS-FON considered, the quality of transmission (QoT) of optical signals is degraded due to the inter-core crosstalk (XT) effect occurring in MCFs. To account for the XT impairment when planning lightpath connections, we make use of a reliable QoT model based on the OSNR estimation, in which the XT impairment is integrated with other physical-layer impairments. To model the lightpath planning optimization problem, we develop a novel mixed-integer programming (MIP) formulation that incorporates the OSNR model. In addition, we propose an efficient heuristic method that is capable of solving larger instances of the optimization problem considered. The results of numerical experiments indicate the low scalability of the MIP method due to the presence of XT and high effectiveness of the heuristic method. The analysis of three different network topologies and two types of MCFs shows a high impact of XT on network performance and limited performance gains from the presence of the central core in a 7-core MCF.

Site Selection Optimization of Reverse Logistics Network for Waste Tires

In order to solve the problem of reverse recycling of waste tires in Heilongjiang Province, this paper chooses the third party as the core to construct a reverse logistics network system of waste tires based on the current situation of reverse logistics of waste tires. The reverse recycling area of waste tires in Heilongjiang Province is divided into five areas (the first area is Harbin; the second area contains Qiqihar, Daqing, and Suihua; the third area contains Yichun, Hegang, Jiamusi, and Shuangyashan; Qitaihe, Jixi, and Mudanjiang form the fourth area; and the fifth area contains Daxinganling and Heihe). The analytic hierarchy process (AHP) is employed to analyze the influence factors of site selection of reverse logistics network and obtain the total weight of each evaluation factor. The weights of economic conditions, the number of waste tires, traffic conditions, development planning and policy, and geographical conditions are obtained as 0.3547, 0.2256, 0.1690, 0.1644, and 0.0863, respectively. The economic conditions and the number of waste tires with high weights are vital factors for selecting a reverse logistics network for waste tires. The linear mixed-integer programming method (LINGO) is utilized for site selection. Harbin, Qiqihar, Daqing, Mudanjiang, Jiamusi, and Suihua are chosen for the recovery center’s site selection, while Daqing, Mudanjiang, and Suihua are chosen as the site selection of remanufacturing enterprise.

Dynamic optimization for coordinated replenishment system considering seasonal demand and price quantity discount

Steel structure (SS), as an essential part of heavy equipment (HE), generally has the characteristics of being heavy, oversized, and with unique shapes, making fabrication, storage, and transportation very costly. Therefore, SS supplier has a strong motivation to coordinate with their customers on all trading activities to obtain an optimal linkage between them to reduce the total relevant cost over the planning horizon. A two-echelon single product replenishment model using mixed integer programming (MIP) was proposed in the past for the SS replenishment problem. However, it cannot easily be used to solve large-size replenishment problems because the computational burden has involved the MIP's branch-and-bound solution process. In this study, the echelon stock concept is adopted to separate the stock holding costs between the supplier and the buyer. Therefore, it can be used to decompose the two-echelon single-item replenishment problem into a series of single-echelon single-item lot-sizing problems. Then, a dynamic programming method is used to solve each single-echelon single-item lot-sizing problem to obtain an optimal solution for the SS replenishment problem in a reasonable time. A numerical example demonstrates that the proposed dynamic programming method is more efficient than the traditional MIP method for the large-size replenishment problem. Finally, sensitivity analysis proves that the proposed method allows the supplier to flexibly and quickly adjust delivery policies in response to changes in demand.

Research on improved artificial bee colony algorithm based on large-scale multi-objective group combination optimization

Large power system contains many generating units, and many factors need to be considered during operation. The unit combination optimization is a nonlinear large-scale optimization problem with multi-objective and multi-constraints, and the existing methods have many shortcomings. Artificial bee colony algorithm has good performance in solving nonlinear optimization problems, but it has some disadvantages such as low optimization efficiency and local extremum. To solve these problems, an improved artificial bee colony algorithm is proposed. The algorithm introduces variable field of vision, adjusts the initial solution and selection strategy, and combines with mutation operation in genetic algorithm. A multi - objective optimization model considering economy and environmental protection is constructed. In order to solve the problem that the calculation time is too long due to the expansion of the unit scale, a phased optimization method is adopted, and the improved algorithm is applied to the start-stop scheduling stage. After determining the start-stop state of the unit, the mixed integer programming method is used for load distribution. For unit contains up to 1000 units of large power grid optimization examples are simulated experiment, the experimental results show that the improved optimization algorithm convergence and global search ability is improved, greatly reduces the computation time of the mass unit combination, multiple objective conditions have also made the ideal results, verify the effectiveness of the proposed method.

Volt/VAR Optimization: A Survey of Classical and Heuristic Optimization Methods

Reactive power optimization and voltage control is one of the most critical components of power system operation, impacting both the economy and security of system operation. It is also one of the most complex optimization problems, being highly nonlinear, and comprising both continuous and discrete decision variables. This paper presents the problem formulation, and a thorough literature review and detailed discussion of the various solution methods that have been applied to the Volt/VAR optimization problem. Each optimization method is described in detail, and its strengths and shortcomings are outlined. The review provides detailed information on classical and heuristic methods that have been applied to the Volt/VAR optimization problem. The classical methods reviewed include (i) first- and second-order gradient-based methods, (ii) Quadratic Programming, (iii) Linear Programming, (iv) Interior-Point Methods, (iv) and mixed-integer programming and decomposition methods. The heuristic methods covered include (i) Genetic Algorithm, (ii) Evolutionary Programming, (iii) Particle Swarm Optimization, (iv) Fuzzy Set Theory, and (v) Expert Systems. A comparative analysis of the key characteristics of the classical and heuristic optimization methods is also presented along with the review.

기존선 고속화에 따른 화물열차 지연영향 분석

According to the plan to speed up the conventional lines, the project to speed up the major rail line is underway. The semi high-speed trains with 250km/h are operating for a passenger from 2021 in the Jungang line. When the speed of the existing railway lines is increased, the track capacity will be lowered due to the speed of difference of each train. Generally, the lowest-speed freight trains have much lower priority than passenger trains, causing serious disruptions in operation. When the 250km/h passenger train is newly operated on the Jungang line, the problem of lowering the priority of freight trains becomes more worse. In this paper, we analyzed the delay effect of freight trains due to the operation of semi-high-speed passenger trains on existing lines. To this end, the trains and stations were represented using a time-space network. And we modeled by mixed integer programming method to solve the problem. The train conflict problem was analyzed through a column generation algorithm. As a result of this study, we suggested the necessity of speeding up freight trains and the appropriate operation speed of freight trains.

Unit commitment of power system with large-scale wind power considering multi time scale flexibility contribution of demand response

The large-scale wind power brings challenges to the power system flexibility. The demand response (DR) can provide flexible resources fast and economical. In this paper, a novel unit commitment model of large-scale wind power system considering the contribution of DR to multi time scale flexibility is presented. According to the multi time scale characteristics of DR resources, the multi time scale scheduling model of DR is established; Considering the flexible resource contribution of DR and the time-varying of wind power forecasting error, the multi time scale flexible resource supply and demand model is developed; Then the unit commitment model of large-scale wind power system considering the multi time scale flexibility contribution of DR resources is proposed. The mixed integer programming method is used as the solver. The proposed model can reflect the balance between flexibility supply and demand for multi time scale, and improve the economy of system operation on the premise of ensuring the system flexibility and security. Case studies are presented to demonstrate the effectiveness of the proposed model.

Examination of Daily Task Constraint in Technician Routing and Scheduling Problem

This paper deals with a multi-term technician routing and programming problem with a different concept. The problem is that technicians with various skills form teams, and these teams perform the tasks that take in distinct locations. Technicians have daily duty capacities while performing tasks. The cases where overtime costs are allowed by combining different technician numbers are analyzed. Paper provides a single-purpose mixed integer programming method for modeling the problem because the model implemented aims to optimize the travel cost while simultaneously minimizing the amount of overwork. The main issue also specifies the division of technicians with different skills into teams, assignment of tasks with different skill requirements to teams, and routes for each team at the same time. The number of customers visited on a route within the daily customer limit. In addition, the model included exceeding the daily customer limit in the model and bound it to a constraint. The various combinations of the daily normal and over-work number of customers were evaluated and examined their effects on the objective function. Finally, the paper presents computational experiments and analyses to evaluate the efficiency of the proposed mathematical formulation and solution approach. The results show that daily task constraints are effective in employee selection and amount of use.