Binary Integer Linear Programming Research Articles

Routing and scheduling of trains and engines in a railway marshalling station yard

In a busy railway marshalling station, train and engine traffic management in the receiving/departure yard plays a crucial role in efficient and stable operations. Traditionally, a track assignment problem (TAP) is solved to assign tracks to trains for berthing at a receiving/departure yard. However, the TAP does not encompass shunting operations in the yard (e.g., engine replacement and train disassembly), which can result in additional scheduling challenges for dispatchers and route conflicts between operations. This paper investigates a train and engine routing and scheduling problem (TERSP) in a receiving/departure yard of railway marshalling stations, which involves simultaneously assigning routes and scheduling route-setting start times for both train and shunting operations to be conducted in the yard. By introducing the concepts of task, activity, and pattern, we transform the original problem into assigning pre-generated patterns incorporating both route and route-setting start time alternatives to activities. The transformed problem is formulated into a compact binary integer linear programming model with a linear number of constraints and the objective of minimizing the total time deviation of all involved tasks. An improved technique that relies on listing all maximal (bi)cliques in a constructed graph is designed to effectively model the time coherence and track section occupation constraints. A heuristic that gradually expands the patterns for the identified key activities by adding more start time alternatives is applied to remedy an infeasible model caused by potential route conflicts. In addition, a rolling horizon algorithm that decomposes the original problem into consecutive smaller stages using either a time-rolling or a train-rolling rule is developed to efficiently solve instances. Finally, numerical experiments based on the physical layouts and real timetables of a receiving yard and a departure yard of a large marshalling station in China are conducted to assess the performance and applicability of our proposed approaches. The results demonstrate that our approaches typically find (near-)optimal solutions within several minutes for the investigated instances by simultaneously addressing different classes of yard operations and resources.

Edge computing offloading strategy for space-air-ground integrated network based on game theory

The limited coverage of terrestrial networks makes it difficult to provide low-latency and high-reliable computing services for users and Internet of Thing (IoT) devices in remote areas such as mountainous regions and oceans. Space-Air-Ground Integrated Network (SAGIN) combined with Mobile Edge Computing (MEC) can provide seamless three-dimensional services for users by deploying edge servers on satellites, Unmanned Aerial Vehicles (UAVs) and ground infrastructures. However, due to the limited heterogeneous computing resources in satellites-UAV clusters network and the energy resources of IoT devices, it brings a significant challenge in determining how to offload the computing tasks generated by ground user devices to satellite edge nodes, UAV edge nodes or locally for processing. In this paper, we first propose a satellites-UAV clusters-ground three-layer edge computing network architecture consisting of a global controller, inter-domain controllers and MEC servers. We then model the task offloading problem as a Binary Integer Linear Programming (BILP) aiming at minimizing the offloading cost composed of delay and energy consumption and prove it is NP-hard. Next, the original offloading problem is transformed to a noncooperative strategic game and the existence of Nash equilibrium is proven using potential games. Finally, we propose a Nash Equilibrium Iteration Offloading algorithm based on Game theory (NEIO-G) to find the optimal offloading strategy. Compared with other baseline algorithms, simulation results demonstrate that the NEIO-G can significantly reduce the system offloading overhead in terms of delay and energy consumption.

Optimization in Territorial Partitioning to Improve the Performance of a Common Building Maintenance Service Contract: A Case Study of a Public Agency in Paraná State, Brazil

All public administration contracts must be evaluated in order to improve their performance while respecting the limits established by laws and regulations. The purpose of this article is to apply an approach to improve the performance of a common building maintenance service contract for the Paraná State Court of Justice (PRCJ), which has a total built area of ​​544 283,79 m² in 224 buildings distributed over 161 counties, by optimizing the territorial partitioning of Paraná State. To partition the state into 14 regions, a binary integer linear programming (BILP) mathematical model is applied to the facilities location problem (FLP) in three scenarios. The results show that Scenario 3 (in which the location of the 14 maintenance offices and the configuration of their areas of activity were optimized) is the best in terms of minimizing the distances traveled by maintenance teams. In this scenario, the total distance traveled would be 9 775 km per day (instead of the current 11 150 km), achieving savings of around 12,3% when compared to the current solution. With this solution, in addition to the distance, the direct and indirect costs associated with the displacement of work teams and the time spent on their corresponding trips would be minimized. Furthermore, the users of maintenance services could be served more quickly, resulting in a higher number of services and greater satisfaction for the target audience of the contract.

Optimization of the pick-and-place sequence of a bimanual collaborative robot in an industrial production line

AbstractThis paper focuses on optimising pick-and-place tasks performed by a dual-arm collaborative robot in a specific shoe manufacturing industry environment. The robot must identify the pieces of a shoe placed on a tray, pick them up, and place them in a shoe mold for further processing. The shoe pieces arrive on the tray in random positions and angles and can be picked up in a different order. Optimising these tasks could increase the assembly speed of each unit and improve shoe production. To achieve this goal, a mathematical model based on binary integer linear programming (BILP) has been developed. This model determines the optimal sequence for picking and placing the shoe pieces in the mold, thus minimising the time required for picking and decision-making. The effectiveness of this approach has been tested using two 3-piece unit shoe models: one for training and another for validation. These models encompass a total of 500 trays. An analysis of the results reveals that BILP offers advantages for task motion planning in complex environments with multiple trajectories and the potential for collisions between arms. The model’s generalizability to shoes with n assembly pieces further confirms its robustness for various piece counts.

Integrated multi-plant collaborative production, inventory, and hub–spoke delivery of make-to-order products

Motivated by make-to-order applications with committed delivery dates in a variety of industries, we investigate the integrated multi-plant collaborative production, inventory, and hub–spoke delivery problem in a complex production–distribution network. This network includes multi-location heterogeneous plants, distribution centers, and customers, for producing customized and splittable orders with one or more general-size multi-type jobs. Completed jobs are transported from plants to distribution centers, and then the orders whose all constituent jobs have arrived are delivered from distribution centers to customer sites. The objective is to make integrated scheduling decisions for production, inventory, and delivery, for minimizing total cost composed of production, transportation, tardiness, and inventory. We first formulate this problem as a mixed-integer programming model, and analyze its intractability by proving that the problem is NP-hard and no approximation algorithms exist with a constant worst-case ratio. We then reformulate this problem as a binary integer linear programming model to select a feasible schedule for each job, and propose a combined column generation and two-layer column enumeration algorithm to solve it. Through extensive numerical experiments, we demonstrate that our proposed algorithm is capable of generating optimal or near-optimal solutions expeditiously and outperforms four benchmark approaches, and gain valuable managerial insights for practitioners.

Analysis of Bus Vulnerability Conducted Using a Synchronized Phasor Measurement Unit in Order to Achieve the Maximum Observability

Phasor measurement units (PMUs) have gained significant interest in recent decades. These instruments are used to measure synchronized phasor data. PMUs are gradually but definitely taking over power grids because of the significant phasor information that they generate for both regular and irregular conditions for the purpose of maintaining safety and control. PMUs may be used for a variety of purposes, including state estimation, which is a common task. In order to make state estimation more reliable, a variety of approaches have been looked into, and one of them is the positioning of PMUs. This paper provides a plan for the implementation of the PMUs, taking into account the potential for failure and vulnerability posed by PMU-equipped buses. Two separate studies were carried out and evaluated with the goal of solving the optimum PMU placement problem (OPPP), which pertains to the grids. The findings of the first study show that the maximum bus observability may be accomplished with the fewest possible number of PMUs, even while taking into consideration the fact that there is a risk that one or more PMUs would malfunction. This investigation was carried out with common measures such as zero injection bus (ZIB) and branch flow measurements, both with and without them, in order to assess the outcomes. The second research focused on selecting the PMU-equipped bus’s vulnerability analysis as its primary area of investigation. All of the tests were completed by using binary integer linear programming. Specifically, the described method is meant to be used with an existing PMU framework and in the case that new locations for new PMUs are necessary to be furnished with existing PMUs. This results confirm that the recommended strategy can be implemented successfully on the IEEE benchmark test systems.

A matheuristic for tri-objective binary integer linear programming

Many real-world optimisation problems involve multiple objectives. When considered concurrently, they give rise to a set of optimal trade-off solutions, also known as efficient solutions. These solutions have the property that neither objective can be improved without deteriorating another objective. Motivated by the success of matheuristics in the single-objective domain, we propose a linear programming-based matheuristic for tri-objective binary integer linear programming. To achieve a high-quality approximation of the optimal set of trade-off solutions, a lower bound set is first obtained using the vector linear programming solver Bensolve. Then, feasibility pump-based ideas in combination with path relinking are applied in novel ways so as to obtain a high-quality upper bound set. Our matheuristic is compared to a recently suggested algorithm that is, to the best of our knowledge, the only existing matheuristic method for tri-objective binary integer linear programming. In an extensive computational study, we show that our method generates a better approximation of the true Pareto front than the state-of-the-art matheuristic on a large set of tri-objective benchmark instances. Since the developed approach starts from a potentially fractional lower bound set, it may also be used as a primal heuristic in the context of linear relaxation-based multi-objective branch-and-bound algorithms.

Linear Programming Models for Leak Detection and Localization in Water Distribution Networks

This study examined the benefits and limitations of formulating linear programming models for water distribution network leak detection and localization considering sequences of demands. First, the consistency of sensitivity matrixes for different demand magnitudes and spatially varying nodal demands was investigated to determine if different sensitivity matrixes are needed for each time step and to determine the demands to use for computing the gradient matrixes. Then several unconstrained and constrained models were developed and their detection and localization performance was evaluated for a network in Austin, Texas. To constrain the failure location in time and space, binary (zero–one) integer linear programming (BILP) was used. A set of threshold-based detection rules was applied to test for the anomalies in the time series, and they were compared for a range of realistic leak sizes. Based on the numerical results, the best optimization model was identified considering multiple detection metrics (detection effectiveness, efficiency, and localization). The best BILP model that constrains leak locations spatially and temporally outperformed the unconstrained model in detecting and locating relatively small leaks.

Near-Optimal Decentralized Diagnosis via Structural Analysis

Health monitoring of current complex systems significantly impacts the total cost of the system. Centralized fault diagnosis architectures are sometimes prohibitive for large-scale interconnected systems, such as distribution systems, telecommunication networks, water distribution networks, or fluid power systems. Confidentiality constraints are also an issue. This article presents a decentralized fault diagnosis method that only requires the knowledge of local models and limited knowledge of their neighboring subsystems. The method, implemented in the decentralized diagnoser design ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$D^{3}$ </tex-math></inline-formula> ) algorithm, is based on structural analysis and can advantageously be applied to high-dimensional systems, linear or nonlinear. Using the concept of isolation on request, a hierarchy is built according to diagnostic objectives. The resulting diagnoser is based on analytical redundancy relations (ARRs) generated along the hierarchy. Their number is optimized via binary integer linear programming (BILP) while still guaranteeing maximal diagnosability at each level. <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$D^{3}$ </tex-math></inline-formula> proves of lower time complexity than its centralized equivalent. It is successfully applied to a nonlinear combined cycle gas-turbine power plant.

A Track-Based Conference Scheduling Problem

The scheduling of conferences is a challenging task that aims at creating successful conference programs that fulfill an often wide variety of requirements. In this work, we focus on the problem of generating conference programs that organize talks into tracks: subevents within the conference that are group-related talks. The main contributions of this work can be organized into three scopes: literature review, problem formulation and benchmarking, and heuristic approach. We provide a literature review of conference scheduling approaches that organizes these approaches within a timetabling problem taxonomy. We also describe the main characteristics of the conference scheduling approaches in the literature and propose a classification scheme for such works. To study the scheduling of conferences that include tracks, we introduce the definition of the track-based conference scheduling problem, a new problem that incorporates tracks in the conference program. We provide a binary integer linear programming model formulation for this problem. Our formulation considers the availability of presenters, chairs, and organizers, the avoidance of parallel tracks, and best paper sessions, among other classical constraints of conference scheduling problems. Additionally, based on our formulation, we propose a simple instance-generation procedure that we apply to generate a set of artificial instances. We complete our work by proposing a heuristic method based on the simulated annealing metaheuristic for solving the track-based conference scheduling problem. We compare the results obtained by our heuristic approach and the Gurobi solver regarding execution time and solution quality. The results show that the proposed heuristic method is a practical approach for tackling the problem as it obtains solutions in a fraction of the time required by Gurobi, while Gurobi is also unable to obtain an optimal solution in the defined time for a subset of the instances. Finally, from a general perspective, this work provides a new conference scheduling problem formulation that can be extended in the future to include other features common in conference programs. Moreover, thanks to the instance generation procedure, this formulation can be used as a benchmark for designing and comparing new solving approaches.

Optimization of gas detectors with the aid of computational fluid dynamics - Imposition of realistic restrictions

Flammable or toxic substances are present during oil extraction, production, and refining. Therefore, control measures, protective barriers, and devices responsible for mitigating consequences are necessary. Flammable or toxic gas detectors play a fundamental role in prevention since they usually have an interface with the emergency shutdown, blowdown, ignition source control, ventilation, alarm system, and firefighting systems. However, technical standards and recommendations do not indicate the exact location or the required number of detectors. Although gas detectors are the most effective devices for gas phase releases, they detect approximately 39% of leaks. Several methodologies have been presented to optimize the position and quantity of the detectors. However, it is not guaranteed that mathematical solutions to the optimization problem can be implemented in practice. The unconstrained optimization result may indicate optimal coordinate sensors in a too high position, requiring cable support and making maintenance and calibration difficult. Also, another possibility is that the detectors are too close to the leak source. This condition could make detection difficult if the leak does not occur precisely in that direction. In order to overcome these difficulties, the present study incorporates distance restrictions from the leak point, distance restrictions from the ground and the equipment side. In other words, the solutions obtained from the optimization are intended to be feasible in the real world for the construction and assembly stage in the field. Furthermore, the present study compared the position of the detectors considering heuristic algorithm and binary integer linear programming, the position of the initial detectors with and without restrictions, spacings between the initial detectors, restrictions of proximity to the leak point, and alarm conditions by 20% and 60% of the lower flammability limit (LFL), which resulted in 60 different optimization scenarios. The main results showed that even with the restrictions imposed, it was possible to reach the minimum number of detectors compared to those without restrictions. All cases were detected when the initial matrix spacing was less than or equal to 1 m, regardless of other restrictions. For comparison, the 5 m spacing, adopted in most studies involving optimization of gas detectors, presented its best result with twice as many devices.

A mixed-integer linear programming model for aggregating multi–criteria decision making methods

Selecting an MCDM method to use in any decision-making problem is always a difficult issue regarding that there is no agreement generally on which method is the most appropriate one. This paper addressed a proposal of a hybrid approach for this problem. Under the assumption that there is no superiority among well-established and accepted MCDM methods, we defined a maximin strategy based on the fact that the lowest correlation between MCDMs and the proposed hybrid approach in terms of alternative rankings should be as high as possible. Even though MCDM methods often rank the alternatives differently, many methods perform similar ranking due to sharing alike mathematical operations. To avoid positive bias towards these methods in an integrated approach, we focused on a prioritizing scheme that supports different rankings. This prioritizing scheme also contributed to hinder the problem of selecting MCDMs with constraining the compound effect of similar rankings. We developed a hybrid decision-making model combining different MCDM methods with prioritizing them by using a binary integer linear programming model. We compared the proposed approach with some well-known prioritizing methods and the results revealed that the proposed approach produced better outcomes in obtaining the desired outputs.

Trajectory Optimization and Phase-Shift Design in IRS-Assisted UAV Network for Smart Railway

The recent trend towards the smart transportation system has spurred the development of the smart railway. However, enabling trains with seamless wireless connectivity using the roadside units (RSUs) is extremely challenging, mostly due to the lack of line of sight link. To address this issue, we propose a novel framework that uses intelligent reflecting surfaces (IRS)-enabled unmanned aerial vehicle (UAV) to provide line of sight communication to the smart railway system. First, we formulate the optimization problem where the objective is to maximize the minimum achievable rate of trains by jointly optimizing the trajectory of UAV and the phase-shift of IRS. Due to the non-convex nature of the formulated problem, it is decomposed into two subproblems: IRS phase-shift problem and UAV trajectory optimization problem. Next, a Binary Integer Linear Programming (BILP) and a Soft Actor-Critic (SAC) are constructed in order to solve our decomposed problems. Finally, comprehensive numerical results are provided in order to show the effectiveness of our proposed framework.

Simple and fast algorithm for binary integer and online linear programming

In this paper, we develop a simple and fast online algorithm for solving a class of binary integer linear programs (LPs) arisen in general resource allocation problem. The algorithm requires only one single pass through the input data and is free of matrix inversion. It can be viewed as both an approximate algorithm for solving binary integer LPs and a fast algorithm for solving online LP problems. The algorithm is inspired by an equivalent form of the dual problem of the relaxed LP and it essentially performs (one-pass) projected stochastic subgradient descent in the dual space. We analyze the algorithm in two different models, stochastic input and random permutation, with minimal technical assumptions on the input data. The algorithm achieves \(O\left( m \sqrt{n}\right) \) expected regret under the stochastic input model and \(O\left( (m+\log n)\sqrt{n}\right) \) expected regret under the random permutation model, and it achieves \(O(m \sqrt{n})\) expected constraint violation under both models, where n is the number of decision variables and m is the number of constraints. The algorithm enjoys the same performance guarantee when generalized to a multi-dimensional LP setting which covers a wider range of applications. In addition, we employ the notion of the permutational Rademacher complexity and derive regret bounds for two earlier online LP algorithms for comparison. Both algorithms improve the regret bound by a factor of \(\sqrt{m}\) by paying more computational cost. Furthermore, we demonstrate how to convert the possibly infeasible solution to a feasible one through a randomized procedure. Numerical experiments illustrate the general applicability and effectiveness of the algorithms.

Security-constrained optimal placement of PMUs using Crow Search Algorithm

For precise power system monitoring, a major focus is on involvement of the latest technology based on phasor measurement units (PMUs). As the sole system monitor, state estimator plays an important role in the security of power system operations. Optimal placement of PMUs (OPP) with numerical observability ensures reliable state estimation. For economical and efficient utilization, there is a need to optimize the placement of PMUs in the power system network. A new approach called Crow Search Algorithm (CSA) devised by others, has been used to solve an OPP problem. The performance of this new approach is compared to the dominant method for an optimization problem — binary integer linear programming (BILP). Comparison studies have also been carried out with particle swarm optimization (PSO) method. The major constraints such as topological, numerical observability conditions with and without zero-injection buses (ZIBs) are considered. Contingencies and limitation of measurement channels in a PMU device are also incorporated as constraints. The main advantage of using the CSA is that it provides multiple location sets for same optimal number of PMUs (optimal number same as obtained by BILP). While BILP method provides only one set of locations for the optimal number of PMUs obtained. This becomes advantageous in planning stage for power engineers for placing PMUs. Test systems considered for the case studies are of varied sizes such as IEEE 14-bus, IEEE 30-bus, IEEE 57-bus and 72-bus practical equivalent system of Indian southern region power grid.

Formulation of Sudoku Puzzle Using Binary Integer Linear Programming and Its Implementation in Julia, Python, and Minizinc

Sudoku is a number puzzle game popular among people with various difficulty levels (easy, medium, hard, and extremely hard). Sudoku can be modeled as a linear programming problem in mathematics, particularly binary integer linear programming (BILP). Completing Sudoku using BILP is quite tricky because it requires many iterations. Therefore, this study aims to analyze the Sudoku problem using the BILP formulation and implement the problem using Julia, Python, and MiniZinc. Out of 15 cases for each difficulty level, Julia performs better than Python and MiniZinc based on computation time. Moreover, Sudoku with easy difficulty levels is solved with a longer computation time than the other three difficulty levels. The computation time for solving BILP is getting faster as the difficulty level of the Sudoku problem increases. This is because Sudoku problems with easy difficulty levels have more known values as clues and generate more constraints than other difficulty levels.

A branch-and-price approach to a variant of the cognitive radio resource allocation problem

Radio-frequency portion of the electromagnetic spectrum is a scarce resource. Cognitive radio (CR) technology has emerged as a promising solution to overcome the spectrum scarcity bottleneck. Through this technology, secondary users (SUs) sense the spectrum opportunities free from primary users (PUs), and opportunistically take advantage of these (temporarily) idle portions, known as spectrum holes. In this correspondence, we consider a variant of the cognitive radio resource allocation problem posed by Martinovic et al. in 2017. The distinguishing feature of this version of the problem is that each SU, due to its hardware limitations, imposes the requirement that the to-be-aggregated spectrum holes cannot be arbitrarily far from each other. We call this restriction as the Maximal Aggregation Range (MAR) constraint, and refer to this variant of the problem as the MAR-constrained hole assignment problem. The problem can be formalized as an NP-hard combinatorial optimization problem. We propose a novel binary integer linear programming (ILP) formulation to the problem. The number of constraints in this formulation is the number of spectrum holes plus the number of SUs. On the other hand, the number of binary decision variables in the formulation can be prohibitively large, as for each legitimate spectrum allocation to each SU, one variable is needed. Due to this difficulty, the resulting 0-1 ILP instances may not even be explicitly describable, let alone be solvable by an ILP solver. We propose a branch-and-price (B&P) framework to tackle this challenge. This framework is in fact a branch-and-bound procedure in which at each node of the search tree, we utilize the so-called (delayed) column generation technique for solving the LP relaxation of the corresponding subproblem. As evidenced by the numerical results, the LP relaxation bounds are very tight. This allows for a very effective pruning of the search space. Compared to the previously suggested formulations, the proposed technique can require much less computational effort.

A Binary Integer Linear Programming Approach for Risk Minimization of a Multi-Mode Resource-Constrained Project Scheduling Problem with Discrete Time-Cost-Quality-Risk Trade-Off

The multi-mode resource-constrained project scheduling problem (MRCPSP) allows project managers to assign schedules and limited resources to project activities while minimizing total duration. This time objective has been extended to include the trade-offs of cost and quality called the discrete time-cost-quality trade-off problem (DTCQTP) to provide project managers an overview of indicators affecting project performance. However, the impacts of the COVID-19 pandemic have led project managers and consultants to assess their projects’ risks regarding scheduling and resource assignment decisions. Hence, this paper aims to extend the MRCPSP and the DTCQTP to include risk at the resource level to highlight the importance of hiring the proper resources in project scheduling. A binary integer linear programming model named the multi-mode resource-constrained project scheduling with discrete time-cost-quality-risk trade-off (MRCPSP-DTCQRT) was developed with risk minimization as the objective function. A case study from prior literature was used to illustrate the model using the open-sourced Python-MIP package, which uses the branch-and-cut methodology for generating optimal solutions. A set of schedules that either prioritize time, cost, quality, risk, or a balance among the four was generated for the use of the project manager to make decisions based on the current situation. Future research may extend the model further to include resource skills, test large-scale case studies, and use other methodologies such as metaheuristics and machine learning to arrive at optimal solutions within reasonable computing time.

Multi-Objective Optimization Aiming to Minimize the Number of Power Quality Monitors and Multiple Fault Estimations in Unbalanced Power Distribution Systems

Utilities are already concerned about power quality monitoring. Several papers proposed strategies to choose the best locations to install power quality monitors, minimizing the investment cost while guaranteeing observability of power quality disturbances, mostly voltage sags. Although several papers studied this problem in power transmission systems, many problems remain unsolved for power distribution systems, such as power quality monitors allocation in unbalanced distribution systems, the impact of distributed generation in allocation methods, and allocation of power quality monitor for fault location. This paper makes contributions in all these three points. It proposes a multi-objective binary integer linear programming model for unbalanced systems, which is still suitable for power transmission systems. Furthermore, we analyze the connection/disconnection of a distributed generator, and our tests showed no need to change allocation regardless of the generator connection status. Finally, the allocation method reduced the multiple estimation problem; therefore, meter-based fault location methods could improve their performance with our allocation.

Optimal PMU Placement in the Presence of Conventional Measurements

This article presents a novel method for optimal phasor measurement unit placement (OPP) for full power system observability in the presence of conventional measurements. The method considers eligible PMU placements that may be omitted by existing OPP methods and hence may provide a better solution and can guarantee that the PMU placement can restore network observability. This is achieved by creating a new observability criterion and a network transformation scheme. The new observability criterion uses injection and zero injection measurements to improve the solution and the network transformation scheme reduces the size of the OPP problem and is a prerequisite for applying the proposed observability criterion. The OPP problem is solved using binary integer linear programming (BILP) or binary integer linear programming (BILP). Therefore, the proposed OPP method can be easily combined with other OPP methods, considering other constraints/scenarios using BILP, such as OPP considering PMU current channel limits. The new method is tested using simulations of the IEEE 14-, 118-, and 2736-bus test systems. These results show that the proposed new method is feasible in terms of execution time, is free of solutions that fail to make the system fully observable, and is capable of identifying optimal placement solutions that may be overlooked by existing OPP methods.