Mixed Integer Linear Programming Solution Research Articles

Multi-fidelity optimization for the day-ahead scheduling of Pumped Hydro Energy Storage

Optimizing the operation of Pumped-Hydro Energy Storage (PHES) requires accurately representing nonlinearities, such as reservoir geometry and water-power conversion efficiency. While traditional methods like Mixed-Integer Linear Programming (MILP) offer theoretical guarantees, they rely on approximations that can lead to suboptimal decisions and costly redispatch or penalties. Because of its inherent approximations, MILP is a low-fidelity optimization model. In this paper, we propose a multi-fidelity approach that combines MILP with a Surrogate-Based Optimization Algorithm (SBOA). MILP solutions are used as warm-starts for the SBOA, which refines the solutions using a high-fidelity simulator of PHES dynamics and redispatch costs. This allows the SBOA to handle nonlinearities and improve the initial MILP solution by exploring areas with higher expected value. Our approach is tested on a PHES unit that participates in the energy and reserve markets in Belgium. The results show that, despite the extensive efforts made in MILP modeling, decisions can still be improved through smart integration with SBOAs.

A new mathematical model and meta-heuristic algorithm for order batching, depot selection, and assignment problem with multiple depots and pickers

The ‘order picking problem’ involves the efficient and organized retrieval of items from shelves to fulfill customer orders. In this study, we consider a new configuration of warehouse system with multiple pickers and depots (m-pickers & n-depots) for manually operated ‘picker-to-parts’ warehouses. The efficiency of the process is measured based on two metrics: i- total order picking distance of each picker ii- total number of pickers assigned to each of depots. The handled problem is called as ‘Order Batching, Depot Selection and Assignment Problem with Multiple Depots and Multiple Pickers (OBDSAPMDMP)11OBDSAPMDMP = Order Batching, Depot Selection, and Assignment Problem with Multiple Depots and Multiple Pickers.’. To solve this complex problem, a new bi-objective Mixed-Integer Linear Programming (MILP) formulation for small-sized problems and a meta-heuristic called ‘Dependent Harmony Search (DHS)22DHS = Dependent Harmony Search.’ for large-sized problems are proposed. The performance of DHS algorithm is evaluated by comparing the optimal results attained by MILP model. For the problem size of 10 orders, the average gap (%) in distance between the solution of DHS and MILP is 4.22%, although in some experiments DHS can find the optimal solution in a very short time. Also, in related analysis, it is seen that constructing multiple depots instead of one left-most located depot decreases total order picking distance by 7.11% on average.

A topology and sizing optimisation method for lightweight sandwich structures subject to dynamic and static constraints

A static-dynamic topology-sizing optimisation method is presented. The solution is based on a sequential Mixed-Integer Linear Programming solution and aims to minimise the mass of a structure subjected to concurrent constraints on static and dynamic response. It is shown that the classical problem of the dynamics of lightweight sandwich structures may be mitigated through core topology and face sheet thickness combinations, retaining the static load carrying capacity while presenting stringent dynamic properties at a low mass penalty.A numerical example, in the form of a load carrying sandwich beam which is excited at different frequencies, is used to demonstrate the method.

A multi-agent learning framework for mixed-integer linear programming

Mixed integer linear programming (MILP) is an important problem in the combinatorial optimization domain, which has wide applications in practical optimization scenarios. Given that most MILP problems fall into the NP-hard category, which the traditional methods may fail to solve, recent research has tried to derive MILP solutions using machine learning techniques. The whole MILP-solving procedure involves lots of modules, such as pre-solving, cut selection, node section, etc., and these modules are closely related and influence each other. However, the previous machine learning-based approaches neglect the connections between these modules, and focus on single-module learning techniques. To address this, we propose an initial step towards a more comprehensive multi-agent learning framework that allows different modules to interact and collaborate. Specifically, our current implementation involves two key modules: HEM for cut selection applied at the root node and GCNN for variable selection. By employing HEM to influence the training of GCNN, these two agents thus work in unison. Through extensive experiments on four MILP datasets in diverse scenarios, we observe significant improvements in solving time and PD integral metrics compared with the state-of-the-art learning-based MILP solving methods. This work lays the groundwork for future development of a fully integrated multi-agent framework.

A hybrid quantum–classical algorithm for mixed-integer optimization in power systems

Mixed Integer Linear Programming (MILP) can be considered the backbone of the modern power system optimization process, with a large application spectrum, from Unit Commitment and Optimal Transmission Switching to verifying Neural Networks for power system applications. The main issue of these formulations is the computational complexity of the solution algorithms, as they are considered NP-Hard problems. Quantum computing has been tested as a potential solution towards reducing the computational burden imposed by these problems, providing promising results, motivating the can be used to speedup the solution of MILPs. In this work, we present a general framework for solving power system optimization problems with a Quantum Computer (QC), which leverages mathematical tools and QCs’ sampling ability to provide accelerated solutions. Our guiding applications are the optimal transmission switching and the verification of neural networks trained to solve a DC Optimal Power Flow. Specifically, using an accelerated version of Benders Decomposition , we split a given MILP into an Integer Master Problem and a linear Subproblem and solve it through a hybrid “quantum–classical” approach, getting the best of both worlds. We provide 2 use cases, and benchmark the developed framework against other classical and hybrid methodologies, to demonstrate the opportunities and challenges of hybrid quantum–classical algorithms for power system mixed integer optimization problems.

Scheduling for additive manufacturing with two-dimensional packing and incompatible items

Additive manufacturing technology can enable low-cost, efficient production of low-demand, highly-complex customized items with reduced lead time if production is judiciously planned. This paper addresses the assignment of requested items into batches and the scheduling of the batches onto 3D printers. The objective is to minimize the manufacturing makespan while satisfying items’ compatibility, two-dimensional no-overlap and containment packing constraints within a batch, and machine’s capability to manufacture a batch. The problem is modeled as a mixed integer linear program (MIP) that solves instances up to 100 items. For hard and large instances, this paper proposes a matheuristic that fathoms packings using a step-wise check procedure. Computational results reveal that the proposed heuristic improves the makespan of MIP solutions of hard instances by 12% on average, with improvements reaching up to 72% for instances with 150 items. They further show that the proposed heuristic finds the best makespan for 88% of all cases. Finally, we provide useful managerial insights for production flexibility and scheduling policies.

Learning From Images: Proactive Caching With Parallel Convolutional Neural Networks

With the continuous trend of data explosion, delivering packets from data servers to end users causes increased stress on both the fronthaul and backhaul traffic of mobile networks. To mitigate this problem, caching popular content closer to the end-users has emerged as an effective method for reducing network congestion and improving user experience. To find the optimal locations for content caching, many conventional approaches construct various Mixed Integer Linear Programming (MILP) models. However, such methods may fail to support online decision making due to the inherent curse of dimensionality. In this paper, a novel framework for proactive caching is proposed. This framework merges model-based optimization with data-driven techniques by transforming an optimization problem into a grayscale image. For parallel training and simple design purposes, the proposed MILP model is first decomposed into a number of sub-problems and, then, Convolutional Neural Networks (CNNs) are trained to predict content caching locations of these sub-problems. Furthermore, since the MILP model decomposition neglects the network resources (such as caching space and link bandwidth) competition among sub-problems, the CNNs' outputs have the risk to be infeasible solutions. Therefore, two algorithms are provided: the first uses predictions from CNNs as an extra constraint to reduce the number of decision variables; the second employs CNNs' outputs to accelerate local search. Numerical results show that the proposed scheme can reduce 71.6% computation time, whose computation time reaches around 28.9 ms, with only 0.8% additional performance cost compared to the MILP solution, which provides high quality decision making in pseudo real-time.

Maximizing Packets Collection in Wireless Powered IoT Networks With Charge-or-Data Time Slots

This paper studies data collection in a wireless powered Internet of things (IoT) network with a hybrid access point (HAP). A fundamental problem at the HAP is to determine the number of time slots over a given planning horizon that is used to charge and collect data from devices. To this end, we outline a mixed integer linear program (MILP) to determine (i) the mode (charge or data) of each slot, (ii) the HAP’s transmit power allocation, and (iii) transmitting devices in data slots. Further, we propose a receding horizon approach whereby the HAP solves the said MILP over a time window using channel estimates from a Gaussian mixture model (GMM). We also outline a data-driven approach. In its offline stage, an IoT network operator first solves the said MILP over an exhaustive collection of channel power gains. The MILP solution for each channel power gain realization is then stored in neural networks. In the online stage, the HAP accesses the trained neural network of each time slot to retrieve its mode. The results show that the rolling horizon and data-driven approach allow the HAP to receive up to 106% and 90% more packets as compared to competing approaches.

Robust Placement and Control of Phase-Shifting Transformers Considering Redispatch Measures

Flexible AC transmission systems (FACTSs) can maximize capacity utilization under time-varying grid usage patterns by actively controlling the power flow of the transmission lines, e.g., with phase-shifting transformers (PST). In this paper, we propose an algorithm to determine the minimum number of PSTs and their location such that the grid can operate robustly for any realization of the (active) power set points from a known, continuous uncertainty set. As we show in our experiments, only considering a few extreme grid scenarios cannot provide this guarantee. The proposed algorithm considers the trade-offs between PST placement and operational decisions, such as PST control and redispatch. By minimizing the worst-case redispatch cost, it yields two affine linear control policies for these as a byproduct. Power flow is modeled as a constrained linear system, and the control design and actuator minimization tasks are formulated as a mixed-integer linear program (MILP). We also design a greedy algorithm, whose optimal value differs less than 20% from the MILP solution while being one to two orders of magnitude faster to compute. The proposed algorithm is evaluated for a small demonstrative 3-bus example and the IEEE 39 bus test system.

IoT-Aerial Base Station Task Offloading With Risk-Sensitive Reinforcement Learning for Smart Agriculture

Aerial base stations (ABSs) allow smart farms to offload processing responsibility of complex tasks from internet of things (IoT) devices to ABSs. IoT devices have limited energy and computing resources, thus it is required to provide an advanced solution for a system that requires the support of ABSs. This paper introduces a novel multi-actor-based risk-sensitive reinforcement learning approach for ABS task scheduling for smart agriculture. The problem is defined as task offloading with a strict condition on completing the IoT tasks before their deadlines. Moreover, the algorithm must also consider the limited energy capacity of the ABSs. The results show that our proposed approach outperforms several heuristics and the classic Q-Learning approach. Furthermore, we provide a mixed integer linear programming solution to determine a lower bound on the performance, and clarify the gap between our risk-sensitive solution and the optimal solution, as well. The comparison proves our extensive simulation results demonstrate that our method is a promising approach for providing a guaranteed task processing services for the IoT tasks in a smart farm, while increasing the hovering time of the ABSs in this farm.

Integrated optimization of demand-driven timetable, train formation plan and rolling stock circulation with variable running times and dwell times

Timetable, train formation plan and rolling stock circulation are the key operational issues for efficient railway operation. The three issues are optimized sequentially based on the order of the operational plan. Sequential optimization manner cannot balance the costs of the three stages effectively and could create an infeasible solution for a later stage when the resources are limited. In addition, because of the computational complexity, multiple variable elements, such as the variable running times, variable dwell times and the coupling/decoupling operations, of a transportation system are ignored. This paper focuses on optimizing timetable, train formation plan and rolling stock circulation simultaneously to minimize costs and meet passenger demand. The key to solving the problem is to determine operation times (i.e., arrival times, departure times, running times and dwell times), the formation type of each train service and rolling stock connections (including the turnaround operations and the coupling/decoupling operations) between these train services. Considering the passenger costs and operator costs, a multi-objective mixed-integer nonlinear programming (MINLP) model is proposed to minimize the total passenger waiting time (TWT), the number of rolling stocks (NR), the number of formations (NF) and the number of coupling/decoupling operations (NC) based on a time–space network. The multi-objective MINLP model is further reformulated into a single-objective mixed-integer linear programming (MILP) model by logical linearizing, piecewise linearizing and fuzzy programming. A numerical example is used in which the exact solution of MINLP obtained by the BARON solver is compared with the approximate solution of MILP obtained by the CPLEX solver to verify the effectiveness of the piecewise linearization. Finally, the model is tested on a real-world case study of the Jinshan Line in Shanghai. The CPLEX solver can efficiently produce the approximate solution within a given computation time in acceptable gaps. The results demonstrate that the integrated model can reduce the number of rolling stocks and improve the utilization rate of rolling stocks. Furthermore, the integrated model can effectively resolve the shortage of rolling stocks when only one depot has rolling stocks or the number of available rolling stocks is limited. In addition, considering the multiple variable elements have a significant effect on the improvement of all the objectives and reduce both the passenger costs and the operator costs.

A Unified MILP Solution Framework for Adaptive Robust Scheduling Problems With Mixed-Integer Recourse Objective

To mitigate the high computational cost of adaptive robust optimization with a mixed-integer recourse objective in real-world power scheduling, a unified mixed-integer linear programming (MILP) solution framework is proposed in this paper. Numerical tests verify the superiority and practicability of the proposed solution framework over the existing decomposition-based solution algorithms.

A Scalable Heuristic for Mission Planning of Mobile Robot Teams

In this work, we investigate a task planning problem for assigning and planning a mobile robot team to jointly perform a kitting application with alternative task locations. To this end, the application is modeled as a Robot Task Scheduling Graph and the planning problem is modeled as a Mixed Integer Linear Program (MILP). We propose a heuristic approach to solve the problem with a practically useful performance in terms of scalability and computation time. The experimental evaluation shows that our heuristic approach is able to find efficient plans, in comparison with both optimal and non-optimal MILP solutions, in a fraction of the planning time.

Three Solution Approaches to Stochastic Multi-Period AC Optimal Power Flow in Active Distribution Systems

This paper proposes three novel solution approaches (A1, A2, A3) to solve stochastic multi-period AC optimal power flow (S-MP-OPF) for day-ahead flexibility procurement from distributed energy resources (DER) in active distribution systems (ADSs). The S-MP-OPF is a mixed-integer nonlinear programming (MINLP) problem due to the binary variables modeling the operation of storage devices and flexible loads. The proposed three approaches have a shared common first step, which resorts to a new mixed-integer linear programming (MILP) model approximation of the S-MP-OPF problem. The MILP model employs second-order Taylor series expansion of trigonometric terms and formulates the linear approximations relying on variables such as square of voltage magnitude and voltage angle difference. This first step serves also the purpose of fixing the binary variables to the values computed by the MILP problem. Then, the approach A1 only checks the AC feasibility of MILP solution while approaches A2 and A3 further optimize continuous variables. Specifically, the sophisticated heuristic approach A2 employs sequential linear programming and AC power flow while the approach A3 models and solves directly the remaining nonlinear programming (NLP) problem. The performances of these approaches, a benchmark MINLP solver, and a state-of-the-art method are thoroughly compared in three radial or weakly meshed ADSs of 34, 31, and 191 nodes, respectively. The numerical results indicate that, albeit the approach A2 performs best overall, these approaches present distinct accuracy vs speed trade-offs, which make them suitable to problems of different sizes and diverse accuracy or speed requirements.

Territory Design for the Multi-Period Vehicle Routing Problem with Time Windows

This study introduces the Territory Design for the Multi-Period Vehicle Routing Problem with Time Windows (TD-MPVRPTW) problem, motivated by a real-world application at a food company’s distribution center. This problem deals with the design of contiguous and compact territories for delivery of orders from a depot to a set of customers, with time windows, over a multi-period planning horizon. Customers and their demands vary over time. The problem is modeled as a mixed-integer linear program (MILP) and solved by a proposed heuristic. The heuristic solutions are compared with the proposed MILP solutions on a set of small artificial instances and the food company’s solutions on a set of real-world instances. Computational results show that the proposed algorithm can yield high-quality solutions within moderate running times. A methodology is proposed in which the territories computed by the proposed heuristic on the past demand of one month are used for the operational routing during the following month, in which the demand is known only one day in advance. An evaluation shows that the territories obtained with our methodology would have led to levels of service significantly better than the ones that were experienced by the company, using a significantly lower number of vehicles to execute the deliveries.

The Product Test Scheduling Problem

This research focused on product test scheduling in the presence of in-process and at-completion inspection constraints. Such testing arises in the context of the manufacture of products that must perform reliably in extreme environmental conditions. Often, these products must receive a certification from prescribed regulatory agencies at the successful completion of a predetermined series of tests. Operational efficiency is enhanced by determining the optimal order and start times of tests so as to minimize the makespan while ensuring that technicians are available when needed to complete in-process and at-completion inspections. We refer to this as the product test scheduling problem. We first formulated a mixed-integer linear programming (MILP) model to identify the optimal solution to this problem and solve it using a commercial optimization package. We also present a genetic algorithm (GA) solution methodology that is implemented and solved in Microsoft Excel. Computational results are presented demonstrating the merits and consistency of the MILP and GA solution approaches across a number of scenarios.

A Branch and Bound Algorithm for Transmission Network Expansion Planning Using Nonconvex Mixed-Integer Nonlinear Programming Models

The branch and bound (BB) algorithm is widely used to obtain the global solution of mixed-integer linear programming (MILP) problems. On the other hand, when the traditional BB structure is directly used to solve nonconvex mixed-integer nonlinear programming (MINLP) problems, it becomes ineffective, mainly due to the nonlinearity and nonconvexity of the feasible region of the problem. This article presents the difficulties and ineffectiveness of the direct use of the traditional BB algorithm for solving nonconvex MINLP problems and proposes the formulation of an efficient BB algorithm for solving this category of problems. The algorithm is formulated taking into account particular aspects of nonconvex MINLP problems, including (i) how to deal with the nonlinear programming (NLP) subproblems, (ii) how to detect the infeasibility of an NLP subproblem, (iii) how to treat the nonconvexity of the problem, and (iv) how to define the fathoming rules. The proposed BB algorithm is used to solve the transmission network expansion planning (TNEP) problem, a classical problem in power systems optimization, and its performance is compared with the performances of off-the-shelf optimization solvers for MINLP problems. The results obtained for four test systems, with different degrees of complexity, indicate that the proposed BB algorithm is effective for solving the TNEP problem with and without considering losses, showing equal or better performance than off-the-shelf optimization solvers.

Investigating the Use of Linear Programming and Evolutionary Algorithms for Multi-Objective Electric Vehicle Charging Problem

With the increasing uptake of electric vehicles (EVs), the need for efficient scheduling of EV charging is increasingly becoming important. A charging station operator needs to identify charging/discharging power of the client EVs over a time horizon while considering multiple objectives, such as operating costs and the peak power drawn from the grid. Evolutionary algorithms (EAs) are a popular choice when faced with problems involving multiple objectives. However, since the objectives and constraints of this problem can be expressed using linear functions, it is also possible to come up with improvised multi-objective formulations which can be solved with exact techniques such as mixed-integer linear programming (MILP). With both approaches having their potential strengths and pitfalls, it is worth investigating their use to inform the algorithmic choices, which the current study aims to address. In doing so, it makes a number of contributions to the topic, including extension of an existing EV charging problem to a multi-objective form; observing some interesting properties of the problem to improve both the MILP and EA solution approaches; and comparing the performance of MILP and EA. The study provides some useful insights into the problem, initial results and quantitative basis for selecting solution approaches, and highlights some areas of further development.

URLLC service-oriented reliable network deployment over MEC enabled TDM-PON based cloud radio access network

New emerging ultra-reliable and low latency communications (URLLC) services have imposed challenged stringent low latency and reliability requirements on fifth generation and beyond (5G/B5G) cloud radio access network (CRAN). However, limited research has been discussed service-oriented reliable network deployment in the mobile edge computing (MEC)-enabled time-division multiplexing-passive optical network (TDM-PON) based CRAN. Moreover, different from most previous protection studies, both optical transport and functional processing resources in multiple domains should be considered in the network deployment of a converged network. To mitigate these limitation, we present a mixed integer linear programming (MILP) for URLLC service-oriented reliable network deployment under the physical and management constraints in MEC-enabled TDM-PON based CRAN. Given the model’s complexity, a heuristic algorithm called “Time-efficient completely isolated protection scheme (TECIPS)” is proposed, which can completely guarantee protection through establishing both primary network topology and strong isolated backup network topology against multiple network failures, while reducing redundant transmission latency and backup recovery latency for URLLC services. Simulation results indicate that our proposed algorithm significantly reduces total latency compared to the benchmark algorithm and receives the near-optimal solution of MILP in both sparse and dense networks with an acceptable time complexity. In addition, the impact of key parameters on the consumption of total transmission latency and backup recovery latency is also evaluated.

Scheduling on flexible flow shop with cost-related objective function considering outsourcing options

This study considers outsourcing decisions in a flexible flow shop scheduling problem, in which each job can be processed by either an in-house production line or outsourced. The selected objective function aims to minimize the weighted sum of tardiness costs, in-house production costs, and outsourcing costs with respect to the jobs due date. The purpose of the problem is to select the jobs that must be processed in-house, schedule processing of the jobs in-house, and finally select and assign other jobs to the subcontractors. We develop a mixed-integer linear programming (MILP) model for the research problem. Regarding the complexity of the research problem, the MILP model cannot be used for large-scale problems. Therefore, four metaheuristic algorithms, including SA, GA, PSO, hybrid PSO-SA, are proposed to solve the problem. Furthermore, some random test problems with different sizes are generated to evaluate the effectiveness of the proposed MILP model and solution approaches. The obtained results demonstrate that the GA can obtain better solutions in comparison to the other algorithms.