Mixed-integer Programming Formulation Research Articles

A mixed-integer programming formulation for optimizing the double row layout problem

The Double Row Layout Problem (DRLP) asks for an arrangement of machines on both sides of a straight line corridor so as to minimize the total cost for transferring materials among machines. The DRLP is NP-Hard and has practical relevance, specially in manufacturing systems design. In this paper, we drastically reduce the time required to solve the problem by constructing a new and effective mixed-integer linear programming (MILP) model of the DRLP. The new model was obtained by reformulating an existing MILP model. This includes tightening some constraints, introducing new variables, implementing constraints to link the new and original variables; and adding valid inequalities and a valid system of equations. To reduce the size of the reformulated model, we eliminate several of the new introduced variables by a substitution using the system of equations. The computational results demonstrate that the proposed model requires considerably smaller computational times compared to the ones in the literature. As a consequence, optimal solutions can now be efficiently found for larger instances of the problem. Previous studies have been able to optimally solve, within reasonable time, instances with size up to 16 machines, while with the new model four instances with 20 machines could be optimally solved.

The home health care routing with heterogeneous electric vehicles and synchronization

AbstractThis paper studies the problem of heterogeneous electric vehicles, fast chargers, and synchronized jobs that have time windows in home healthcare routing and scheduling. We consider a problem that aims to establish daily routes and schedules for healthcare nurses to provide a variety of services to patients located in a scattered area. Each nurse should be assigned to an electric vehicle (EV) from a heterogeneous fleet of EVs to perform the assigned jobs within working hours. We consider three different types of EVs in terms of battery capacity and energy consumption. We aim to minimize the total cost of energy consumption, fixed nurse cost, and costs arising from the patients that cannot be served within the working day. We model the problem as a mixed integer programming formulation. We develop a hybrid metaheuristic based on a greedy random adaptive search procedure heuristic, to generate good quality initial solutions quickly, and an adaptive variable neighborhood search algorithm to generate high quality solutions in reasonable time. The hybrid metaheuristic employs a set of new advanced efficient procedures designed to handle the complex structure of the problem. Through extensive computational experiments, the performance of the mathematical model and the hybrid metaheuristic are evaluated. We conduct analyses on the robustness of the metaheuristic and the performance contribution of employing adaptive probabilities. We analyze the impact of problem parameters such as competency requirements, job duration, and synchronized jobs.

The Vehicle Routing Problem with Cross-Docking and Scheduling at the Docking Station: Compact formulation and a General Variable Neighborhood Search metaheuristic

In this article, we introduce the Vehicle Routing Problem with Cross-Docking and Scheduling at the Docking Station (VRPCDSD), which integrates pickup and delivery routing decisions with the cross-docking scheduling of the unloading and reloading of requests. We present a compact Mixed Integer Programming (MIP) formulation and a General Variable Neighborhood Search (GVNS) metaheuristic for the problem. The proposed GVNS employs several routing and scheduling neighborhoods that require problem-specific adjustments due to the integrated characteristic of the problem. Computational experiments indicate that the MIP formulation can solve instances with up to 31 nodes (15 requests) within the specified time limit when using a standard commercial solver. Furthermore, the GVNS metaheuristic can obtain high-quality solutions that at least match those encountered by the solver for the tested instances, achieving considerably large improvements over the latter especially as the number of requests increases. In fact, GVNS is able to find the optimal solution for all the instances for which the optimal is known. In addition, GVNS is a viable alternative for tackling larger instances of VRPCDSD and is very robust in that it obtains low deviations from the best-encountered solutions when multiple executions are performed. We also present some insights on how the different cross-docking operation costs impact the optimal solutions.

Deep learning enhanced mixed integer optimization: Learning to reduce model dimensionality

This work introduces a framework to address the computational complexity inherent in Mixed-Integer Programming (MIP) models by harnessing the potential of deep learning. By employing deep learning, we construct problem-specific heuristics that identify and exploit common structures across MIP instances. We train deep learning models to estimate complicating binary variables for target MIP problem instances. The resulting reduced MIP models are solved using standard off-the-shelf solvers. We present an algorithm for generating synthetic data enhancing the robustness and generalizability of our models across diverse MIP instances. We compare the effectiveness of (a) feed-forward neural networks (ANN) and (b) convolutional neural networks (CNN). To enhance the framework's performance, we employ Bayesian optimization for hyperparameter tuning, aiming to maximize the occurrence of global optimum solutions. We apply this framework to a flow-based facility location allocation MIP formulation that describes long-term investment planning and medium-term tactical scheduling in a personalized medicine supply chain.

Research on airport apron planning strategy in emergency situations

In events of local war or highly infectious epidemics, airports face the problem of how to park numerous non-missioned aircraft on aprons. In this study, the apron planning problem in emergency situations (APPES) was considered to replan existing parking stands and taxiways, thus determining the numbers and locations of different aircraft types. A mixed-integer programming formulation for the APPES was developed to maximise the total revenue for the apron, where the aircraft was planned to be parked based on factors such as the safety distance and aircraft taxiing method. A two-stage approach based on the discrete particle swarm optimisation algorithm and bottom-left algorithm was designed to solve the APPES. Random instances of different sizes were generated, and the experimental results demonstrated the effectiveness of the proposed approach. The two-stage approach could find optimal solutions to the APPES for 19 of the 20 small instances and could well approximate optimal solutions for the medium and large instances within a reasonable timeframe. Parking priority factor and aircraft taxiing methods sensitivity analyses were also designed to develop a strategy that can adhere to the preferences of decision makers or increase the total revenue.

Benders decomposition algorithms for minimizing the spread of harmful contagions in networks

The COVID-19 pandemic has been a recent example for the spread of a harmful contagion in large populations. Moreover, the spread of harmful contagions is not only restricted to an infectious disease, but is also relevant to computer viruses and malware in computer networks. Furthermore, the spread of fake news and propaganda in online social networks is also of major concern. In this study, we introduce the measure-based spread minimization problem (MBSMP), which can help policy makers in minimizing the spread of harmful contagions in large networks. We develop exact solution methods based on branch-and-Benders-cut algorithms that make use of the application of Benders decomposition method to two different mixed-integer programming formulations of the MBSMP: an arc-based formulation and a path-based formulation. We show that for both formulations the Benders optimality cuts can be generated using a combinatorial procedure rather than solving the dual subproblems using linear programming. Additional improvements such as using scenario-dependent extended seed sets, initial cuts, and a starting heuristic are also incorporated into our branch-and-Benders-cut algorithms. We investigate the contribution of various components of the solution algorithms to the performance on the basis of computational results obtained on a set of instances derived from existing ones in the literature.

Solving the three-dimensional open-dimension rectangular packing problem: A constraint programming model

This paper addresses the three-dimensional open-dimension rectangular packing problem (3D-ODRPP). This problem addresses a set of rectangular boxes of given dimensions and a rectangular container of open dimensions. The objective is to pack all boxes orthogonally into the container while minimizing the container volume. Real-world applications of the 3D-ODRPP arise in production systems with operations of shipping or moving. The literature has presented mainly mixed-integer programming (MIP) formulations and their linearization techniques for the problem allied with general-purpose optimization solvers. To model and solve the 3D-ODRPP, we propose a constraint programming model based on a position-free modeling approach with logic operators. We ran computational experiments to assess the performance of the proposed model compared to the benchmark MIP models from instances of the literature. The results show our approach is competitive in different sets of problem instances in terms of reaching optimality as well as providing satisfactory feasible solutions quickly.

Collaborative Scheduling Optimization of Container Port Berths and Cranes under Low-Carbon Environment

Motivated by the need for a green and low-carbon economy, we explore the co-scheduling optimization of berths and cranes. Our aim is to balance the carbon tax and operating costs of ports under uncertain conditions, proposing an innovative nonlinear mixed-integer programming formulation. To address this optimization challenge, we have developed an enhanced version of the adaptive spiral flying dung beetle algorithm (ASFDBO). In order to evaluate the performance of the ASFDBO algorithm, we performed a benchmark function test and a convergence analysis with other recognized metaheuristics. In addition, we verified the practical applicability of the ASFDBO algorithm in different test scenarios. Through numerical experiments, we analyze the feasibility and effectiveness of the algorithm’s scheduling solutions and improvement strategies. Results indicate that our collaborative scheduling optimization, which considers both carbon and production costs, achieves feasible solutions and reduces carbon expenses. Finally, we investigate the impact of different carbon tax rates on the joint scheduling optimization of berths and quay cranes, and the results show that a reasonable carbon tax policy can effectively reduce the carbon emissions of ports.

Multi-objective mammography unit location–allocation problem: A case study

This work addresses the Multi-Objective Mammography Unit Location–allocation Problem (MOMULAP), aiming to meet three objectives: maximize mammography screening coverage, minimize the total distance traveled weighted by the number of users, and maximize equity in access to mammography screening. We introduce a mixed-integer nonlinear programming (MINLP) formulation to represent the MOMULAP and algorithms based on the Non-dominated Sorting Genetic Algorithm II (NSGA-II) and Strength Pareto Evolutionary Algorithm (SPEA2) for treating it. The algorithms were tested with data from seven Brazilian states. In these states, the number of cities ranges from 139 to 853, equipment from 23 to 347 units, and estimated annual demand for screenings from 96,592 to 1,739,085. The solutions provided by this work allow health managers to choose the appropriate location and allocation of the mammography units, considering different objectives.

First passage time interdiction in Markov chains

We introduce a new variant of the network interdiction problem with a Markovian evader that randomly chooses a neighboring vertex in each step to build their path from designated source(s) to terminal(s). The interdictor's goal is to maximize the evader's minimum expected first passage time. We establish sufficient conditions for the interdiction to not be counter-productive, prove that the problem is NP-hard, and demonstrate the model's usefulness by solving a mixed-integer programming formulation on a test bed of social networks.

General VNS for asymmetric vehicle routing problem with time and capacity constraints

We consider a real-life transportation problem from a hypermarket company in Serbia that consists of delivering online ordered and possibly perishable goods from a single depot to multiple customers. It can be modeled as an asymmetric vehicle routing problem that requires visiting and serving all customers using a limited number of homogeneous vehicles and taking into account time and capacity constraints. The objective function to be minimized is the total distance traveled. We provide a mixed-integer programming (MIP) formulation of the problem and develop several Local Search based metaheuristic methods exploring combinatorial formulation of the considered problem: Multistart Local Search (MLS), Greedy Randomized Adaptive Search Procedure (GRASP), and several variants of General Variable Neighborhood Search (GVNS) methods. All methods are compared on real-life instances provided by the hypermarket company and benchmark instances from the relevant literature. The obtained results show the superiority of GVNS-based methods with respect to both solution quality and running time, confirming that systematic search procedures have more success when dealing with hard optimization problems.

Mixed Integer Programming based Placement Refinement by RSMT Model with Movable Pins

Placement is a critical step in the physical design for digital application specific integrated circuits (ASICs), as it can directly affect the design qualities such as wirelength and timing. For many domain specific designs, the demands for high performance parallel computing result in repetitive hardware instances, such as the processing elements in the neural network accelerators. As these instances can dominate the area of the designs, the runtime of the complete design’s placement can be traded for optimizing and reusing one instance’s placement to achieve higher quality. Therefore, this work proposes a mixed integer programming (MIP)-based placement refinement algorithm for the repetitive instances. By efficiently modeling the rectilinear steiner tree wirelength, the placement can be precisely refined for better quality. Besides, the MIP formulations for timing-driven placement are proposed. A theoretical proof is then provided to show the correctness of the proposed wirelength model. For the instances in various popular fields, the experiments show that given the placement from the commercial placers, the proposed algorithm can perform further placement refinement to reduce 3.76%/3.64% detailed routing wirelength and 1.68%/2.42% critical path delay under wirelength/timing-driven mode, respectively, and also outperforms the state-of-the-art previous work.

A Spatial Optimization Model for Delineating Metropolitan Areas

A metropolitan area comprises a collection of cities and counties bound by strong socioeconomic ties. Despite the pivotal role that metropolitan areas play in regional economics, their delineation remains a challenging task for researchers and urban planners. Current threshold-based delineation methods select counties based on their connection strength with prespecified core counties. Such an approach often neglects potential interactions among outlying counties and fails to identify polycentric urban structures. The delineation of a metropolitan area is fundamentally a spatial optimization problem, whose objective is to identify a set of counties with high interconnectivity while also meeting specific constraints, such as area, contiguity, and shape. In this study, we present a novel spatial optimization model designed for metropolitan area delineation. This model aims to maximize intercounty connection strength in terms of both industry and daily life. This approach ensures a more accurate representation of the multicore structure that is commonly seen in developed metropolitan areas. Additionally, our model avoids the possibility of holes in metropolitan area delineation, leading to more coherent and logical metropolitan boundaries. We provide a mixed-integer programming formulation for the proposed model. Its efficacy is demonstrated by delineating the boundaries of the Nanjing and Lhasa metropolitan areas. This study also delves into discussions and policy implications pertinent to both of these metropolitan areas.

Mathematical programming formulations for the reclaimer scheduling problem with sequence-dependent setup times and availability constraints

In the context of Industry 4.0, which encompasses advanced technologies and interconnected systems, the integration of optimization techniques assumes a crucial role in addressing resource management challenges across various sectors, including manufacturing systems. In bulk ports, which are an integral part of the manufacturing and logistics chain, one of the most critical resource management challenges is the scheduling of automated or semi-automated reclaimers. These machines are responsible for reclaiming dry bulk materials for loading onto vessels using ship-loaders. The efficient operation of reclaimers directly impacts the overall efficiency of the port. However, the current works for the reclaimer scheduling problem (RSP) overlook the necessity of maintenance activities and assume continuous machine availability. Nonetheless, the inclusion of preventive maintenance activities is critical to ensuring optimal scheduling decisions that minimize production disruptions and downtime. To address these challenges, this paper proposes a novel approach that considers periodic preventive maintenance activities of the reclaimers. We consider two cases to tackle this problem. The first case involves the optimization of material handling within a system of two stockpads and one reclaimer machine. For this case, two novel Mixed Integer Programming (MIP) formulations are developed to drive efficient scheduling decisions and maximize productivity. Furthermore, for the second case representing a more complex system comprising three stockpads and two reclaimers, a novel MIP formulation is devised. The effectiveness of the proposed mathematical formulations is rigorously tested through the use of randomly generated instances based on a real-world case.

Minimizing File Transfer Time in Opportunistic Spectrum Access Model

We study the file transfer problem in opportunistic spectrum access (OSA) model, which has been widely studied in throughput-oriented applications for max-throughput strategies and in delay-related works that commonly assume identical channel rates and fixed file sizes. Our work explicitly considers minimizing the file transfer time for a given file in a set of heterogeneous-rate Bernoulli channels, showing that max-throughput policy doesn't minimize file transfer time in general. We formulate a mathematical framework for static extend to dynamic policies by mapping our file transfer problem to a stochastic shortest path problem. We analyze the performance of our proposed static and dynamic optimal policies over the max-throughput policy. We propose a mixed-integer programming formulation as an efficient alternative way to obtain the dynamic optimal policy and show a huge reduction in computation time. Then, we propose a heuristic policy that takes into account the performance-complexity tradeoff and consider the online implementation with unknown channel parameters. Furthermore, we present numerical simulations to support our analytical results and discuss the effect of switching delay on different policies. Finally, we extend the file transfer problem to Markovian channels and demonstrate the impact of the correlation of each channel.

Modeling and solving the fuel distribution problem with unloading precedence and loading sequence considerations

AbstractThis paper presents a real-world liquid fuel distribution problem involving a heterogeneous fleet of multi-compartment vehicles servicing a set of orders of different fuels. Two new features are introduced that affect significantly the performance of the fuel distribution process in terms of safety and efficiency: (i) loading trucks so that the payload remains balanced throughout each phase of a delivery route, and (ii) sequencing the requests for loading trucks at the depot loading facilities. A Mixed Integer Programming formulation is presented and an Adaptive Large Neighbourhood Search algorithm with various novel features is developed and benchmarked. A new loading model is formed and solved for allocating the ordered items (fuel) to vehicle compartments, as part of constructing/repairing delivery routes. The computational performance of the proposed solution approach has been tested on a series of benchmark problems. Moreover, a series of experiments were performed in order to assess the effect of the balanced loading constraints on the traveled distance. The results indicate that the effect of this type of constraint on the total traveled distance is kept at a reasonable level, reaching a maximum increase of 4.37%. The computational tools presented in this work may accommodate the dispatchers’ work in producing efficient and safe delivery routes while managing efficiently any potential bottleneck in the truck loading facilities.

Modeling and optimization of steady flow of natural gas and hydrogen mixtures in pipeline networks

We extend the canonical problems of simulation and optimization of steady-state gas flows in pipeline networks with compressors to the transport of mixtures of highly heterogeneous gases injected throughout a network. Our study is motivated by proposed projects to blend hydrogen generated using clean energy into existing natural gas pipeline systems as part of efforts to reduce the reliance of energy systems on fossil fuels. Flow in a pipe is related to endpoint pressures by a basic Weymouth equation model, with an ideal gas equation of state, where the wave speed depends on the hydrogen concentration. At vertices, in addition to mass balance, we also consider mixing of incoming flows of varying hydrogen concentrations. The problems of interest are the heterogeneous gas flow simulation (HGFS), which determines system pressures and flows given fixed boundary conditions and compressor settings, as well as the heterogeneous gas flow optimization (HGFO), which extremizes an objective by determining optimal boundary conditions and compressor settings. We examine conditions for uniqueness of solutions to the HGFS, as well as compare and contrast mixed-integer and continuous nonlinear programming formulations for the HGFO. We develop computational methods to solve both problems, and examine their performance using four test networks of increasing complexity.

Towards a circular economy with waste‐to‐resource system optimization

AbstractMotivated by mounting pressures to achieve environmental sustainability, and the emergence of online waste exchange platforms, in this work we propose an optimization‐based framework for studying waste‐to‐resource formation in the industry. We first develop a linear programming market clearing model comprising self‐interested agents participating in waste‐to‐resource trading. We then embed this in a bi‐level capacity optimization problem under uncertainties in the agents' reserve prices, the goal of which is to maximize the economic savings achieved by waste‐to‐resource. We show that the resulting robust optimization models admit linear mixed integer programming formulations that can be readily computed using available solvers. Furthermore, we provide the formulas for applying an equitable levy and incentive scheme, to enable capacity cost‐sharing among the agents. Numerical studies based on a case of organic waste streams demonstrate the usefulness of the proposed models in generating valuable insights for decision support in the design of waste‐to‐resource markets.

Sharing in construction projects — On determining optimal container assignments for the on-site accommodation of trades

The sharing economy is among the top trending topics in operations research. In this paper, we introduce a novel sharing application encountered in the construction industry. The problem under research, which has been developed in close cooperation with an industrial partner, is concerned with the on-site accommodation of trades in large-scale construction projects. In such projects, a construction logistics company provides containers gathered in an area of the site and assigns trades to them based on their requests issued online. Usually, a single trade will only remain on-site for a certain period of time, i.e. for as long as it is carrying out its activities in the project. Similarly, containers will only be provided for a certain time interval. This dynamic nature may require re-assigning trades as the project progresses. As such re-assignments impose costs for the logistics company, it is the goal to keep them to a minimum. We formally define the resulting optimization problem, present two mixed-integer programming formulations and develop a fix-and-optimize heuristic. In an extensive computational study, we show the latter to not only clearly outperform standard solver CPLEX operating on the holistic mixed-integer program, but to also provide substantial savings over the assignment policy currently applied in practice.

A novel framework for the day-ahead market clearing process featuring the participation of distribution system operators and a hybrid pricing mechanism

The electricity market liberalization propelled by the increased operational flexibility observed in distribution systems urges the development of mathematical models for competitive electricity markets accounting for the participation of distribution systems operators. Most existing approaches model the interaction between the agents and the market as a bilevel programming problem, wherein the market is characterized at the lower level. Hence, the upper-level agent’s objective function is favored over minimizing the costs of supplying the system’s demand. In this paper, a new bilevel formulation is proposed. In the upper level, the market operator determines the optimal energy price that ensures the demand supply and maximizes the overall social welfare. In the lower level, the generation companies and distribution systems operators determine the amount of energy injected/extracted into/from the transmission system to maximize profits/minimize operational costs. A new pricing method is proposed in this paper as an alternative to the transmission system’s dual variables, which are inaccessible under the proposed framework. The resulting nonlinear mixed-integer bilevel programming formulation is transformed into an equivalent single-level mixed-integer linear program. Numerical results illustrate the proposed approach’s effectiveness in maintaining the minimum operational costs while distributing the payoff amongst the agents.