Problem Size Reduction Research Articles

Problem size reduction methods for large CVRPs

We solve the Capacitated Vehicle Routing Problem (CVRP) by introducing a novel approach to problem size reduction. We propose the generation of short sequences of nodes called “sections”, which effectively act as single nodes in a reduced CVRP that is faster and easier to solve. Three section generation methods are compared, and the trade-off between solution quality and computation time savings is evaluated. We show that reduced problem sizes of up to around 60 percent of the original problem size, result in only modest decreases in solution quality, but allow for significant reductions of computation time, regardless of the optimization algorithm used. Our findings highlight the potential benefits of problem aggregation and size reduction for large-scale CVRPs and suggest opportunities for further improving aggregation methods.

Scalable Combinatorial Vertiport Placement Method for an Urban Air Mobility Commuting Service

Vertiport locations have a significant impact on the time savings provided by an urban air mobility (UAM) commuting service relative to ground transportation and, thereby, greatly affect the value proposition and demand for UAM commuting services. In this paper, we present a discrete combinatorial vertiport placement method with a flexible objective function capable of directly optimizing for commuting demand. Preprocessing and postprocessing formulations that effectively reduce problem size and increase solution quality are also described. Demand-maximizing vertiport placement results are provided for Atlanta, New York City, San Francisco, and Seattle. We also present results showing the sensitivity of potential commuting demand to number of vertiports, ticket price, and service delay. Additionally, results that illustrate the impact of limiting vertiport placement to locations with existing airport and helipad infrastructure are presented. Finally, vertiport placement results optimized using a capacitated profitability-maximization objective are outlined.

Effectiveness of the MineReduce approach in reducing the size of combinatorial optimization problem instances

Previous work has shown that the performance of metaheuristics can benefit from using data mining techniques, which can improve the obtained solutions. In a strategy that has been successfully used for over a decade, data mining techniques are applied to extract patterns from good solutions found in the early stages of the heuristic process, and these patterns are introduced into the solutions generated afterwards. Recently, a novel approach that uses data mining for problem size reduction, called MineReduce, has been proposed and achieved even more impressive results in improving metaheuristics. In this work, we apply the MineReduce approach to improve the performance of a multi-start iterated tabu search algorithm. The results show that with the incorporation of the MineReduce approach, the method can obtain better solutions while spending less time. Additionally, we assessed the effectiveness of the size reduction performed by MineReduce, comparing it to a kernelization algorithm. Despite the lack of guarantees on optimality or size-bounding, the reduction carried out by MineReduce was effective in practice.

Efficient Approximation Methods for Lexicographic Max-Min Optimization

Lexicographic max-min (LMM) optimization is of considerable importance in many fairness-oriented applications. LMM problems can be reformulated in a way that allows to solve them by applying the standard lexicographic maximization algorithm. However, the reformulation introduces a large number of auxiliary variables and linear constraints, making the process computationally complex. In this paper, two approximation schemes for such a reformulation are presented, resulting in problem size reduction and significant performance gains. Their influence on the quality of the solution is shown in a series of computational experiments concerned with the fair network dimensioning and bandwidth allocation problem.

Running the Dual-PQC GAN on noisy simulators and real quantum hardware

In an earlier work [1], we introduced dual-Parameterized Quantum Circuit (PQC) Generative Adversarial Networks (GAN), an advanced prototype of quantum GAN. We applied the model on a realistic High-Energy Physics (HEP) use case: the exact theoretical simulation of a calorimeter response with a reduced problem size. This paper explores the dual-PQC GAN for a more practical usage by testing its performance in the presence of different types of quantum noise, which are the major obstacles to overcome for successful deployment using near-term quantum devices. The results propose the possibility of running the model on current real hardware, but improvements are still required in some areas.

The Cauchy problem for the 3D Poisson equation: Landweber iteration vs. horizontally diagonalize and fit method

Abstract The horizontally diagonalize and fit (HDF) method is proposed to solve the ill-posed Cauchy problem for the three-dimensional Poisson equation with data given on the part of the boundary (a continuation problem). The HDF method consists in discretization over horizontal variables and transformation of the system of differential equations to a diagonal form. This allows to uncouple the original three-dimensional continuation problem into a moderate number of one-dimensional problems in the vertical dimension. The problem size reduction can be carried taking into account the noise level, so that the number k of one-dimensional problems appears to be a regularization parameter. Our experiments show that HDF is applicable to large-scale problems and for n ≤ 2500 {n\leq 2500} is significantly more efficient than Landweber iteration.

A relax-and-fix Pareto-based algorithm for a bi-objective vaccine distribution network considering a mix-and-match strategy in pandemics

Mass vaccination in pandemics is one of the most effective strategies to halt disease transmission. A well-organized vaccination plan is essential for guaranteeing its success in cases with limited vaccines. On the other hand, most vaccines need more than one dose for appropriate immunity, intensifying the vaccine shortage probability and complicating decision-making. This study presents a bi-objective mixed-integer linear model for a vaccine distribution chain problem, simultaneously considering economic and social objectives, with multi-dose vaccination adaptable even for booster doses. Moreover, the compatible vaccines in each vaccination process are considered a mix-and-match strategy to diversify vaccination alternatives and alleviate the vaccine shortage risk. It is shown that the problem is NP-hard, and we develop two heuristic and meta-heuristic algorithms to solve real-size instances in a reasonable time. Both algorithms use problem size reduction and problem decomposition techniques to produce an approximate Pareto front for the problem. The heuristic algorithm employs relax-and-fix and fix-and-optimize techniques, and an epsilon-constraint method to solve the multi-objective problem. The proposed meta-heuristic algorithm is based on the hybrid of two well-known evolutionary algorithms, particle swarm optimization, and genetic algorithm. It also uses a multi-objective framework called PESA-II to generate the Pareto front. Both proposed heuristic and meta-heuristic algorithms can employ parallel computing, which significantly reduces computational time. Finally, we investigate the proposed algorithms’ performance using 30 random problem instances and a case study from Iran. The results demonstrate that heuristic and meta-heuristic algorithms have obtained a reasonable solution with a maximum of 6.9% and 16.3% cost objective function gap respectively. The heuristic algorithm is superior to the meta-heuristic in terms of Pareto front quality metrics.

A cost-technical profit-sharing approach for optimal energy management of a multi-microgrid distribution system

- The primary purpose of this paper is to perform optimal energy management in a distribution network (DN) consisting of several microgrids (MG). A cooperative game method has been used to achieve the best coalitions to diminish the total system costs. A fair profit-sharing procedure is defined as the individual contribution value (ICV) to share profits between players. In addition to ICV, as an economic indicator, the DN's maximum voltage deviation (MVD) in each game has been used to allocate rewards to each game's players (MGs). Studies have been executed on a modified 33-bus DN consisting of three distinct MGs. In the system under study, there is no direct connection among MGs. The case study also deliberates the uncertainties of RESs and technical real power losses in the DN considering demand response program. The paper results concede that despite the reduction in the costs of MMGs due to the application of DRP, it is still possible to reduce the total costs of an MMG system by forming coalitions and adopting proper procedures to profit allocation. In addition, the proposed profit-sharing method based on ICVs and MVDs is simple, consistent, and clear, thus facilitating calculations. Examining the games one by one also identifies the inappropriate games/players to form a coalition and helps reduce problem size in the upcoming scheduling. Solutions obtained through the combined use of ICV and MVD indicators are more profitable, have better technical characteristics, and lead to larger alliances with more MGs participation, increasing network flexibility.

Minimizing a linear objective function under a max-overlap function fuzzy relational equation constraint

Max-t-norm compositions are commonly utilized to optimize a linear objective function subject to fuzzy relational equations, especially, max-min and max-product compositions. However, the associativity is not forcefully needed in many cases. In this paper, the max-overlap function composition is considered for the same optimization model. Then some properties of the solution set are obtained. According to these properties, the characterization of the optimal solution for the optimization model is proposed. Furthermore, a simple value matrix with rules is proposed to reduce problem size. Thus, a solution procedure is presented for determining optimal solutions without translating such an optimization problem into two sub-problems. A numerical example is provided to illustrate the procedure.

Introducing preferences in scheduling applications

In some applications like fabric dying, the sequence in which jobs are processed on a single machine can influence product quality, i.e., there is a strong preference to run a certain type of job after another for reasons that do not influence regular scheduling objectives like total completion time. These reasons include minimizing changes in color families on machines (schedule preferences) and reducing the difference in shades between consecutive jobs (shade inconsistency). Nonetheless, the scheduling of jobs on machines must consider sequence-dependent setup times as well. In this paper, a mixed integer linear program has been presented to schedule jobs and account for setup times. The scheduling preferences and shade differences have been modeled as two new objectives, along with traditionally used objectives such as makespan, lateness and total setup time. The MILP is found to be useful only for small problems. To tackle larger problems, several heuristic methods are proposed. In order to obtain solutions quickly, multiple construction heuristics are developed based on SPT/LPT/EDD/LFJ rules and tested on a variety of industrially-inspired data sets. To produce schedules that perform well across multiple objectives at the same time, three additional tailored heuristics are developed. Both the original and tailored heuristics are compared to the optimal solutions for small-sized problems. In addition, a problem size reduction heuristic has been developed which can segment the larger MILP into smaller independently solvable MILPs by reducing the possible machines on which a job can be processed. For these larger problems, the heuristics are compared among themselves as well as against the problem size reduction heuristic.

Numerical Study of Soil-Thawing Effect of Composite Piles Using GMsFEM

During construction works, it is advisable to prevent strong thawing and an increase in the moisture content of the foundations of engineering structures in the summer. Since the density of water and ice differ, due to the difference bulging of the foundation sections can occur when it freezes back in winter. In this work, the effect of fiber-reinforced piles on the thermal field of the surrounding soil is investigated numerically; that is, the study of the influence of aggregates with high and low thermal-physical properties on the temperature of frozen soils is conducted. Basalt and steel fiber reinforcement are compared. The difficulty of this work is that the inclusions inside piles are too small compared to the pile itself. Therefore, to solve the Stefan problem, a generalized multiscale finite element method (GMsFEM) was used. In the GMsFEM, the usual conforming partition of the domain into a coarse grid was used. It allowed reducing problem size and, consequently, accelerating the calculations. Results of the multiscale solution were compared with fine-scale solution, the accuracy of GMsFEM was investigated, and the optimal solution parameters were defined. Therefore, GMsFEM was shown to be well suited for the designated task. Collation of basalt and steel fiber reinforcement showed a beneficial effect of high thermal conductive material inclusion on freezing of piles in winter.

Shipping Domain Knowledge Informed Prediction and Optimization in Port State Control

Maritime transportation is the backbone of global supply chain. To improve maritime safety, protect the marine environment, and set out seafarers’ rights, port state control (PSC) empowers ports to inspect foreign visiting ships to verify them comply with various international conventions. One critical issue faced by the port states is how to optimally allocate the limited inspection resources for inspecting the visiting ships. To address this issue, this study first develops a state-of-the-art XGBoost model to accurately predict ship deficiency number considering ship generic factors, dynamic factors, and inspection historical factors. Particularly, the XGBoost model takes shipping domain knowledge regarding ship flag, recognized organization, and company performance into account to improve model performance and prediction fairness (e.g., for two ships that are different only in their flag performances, the one with a better flag performance should be predicted to have a better condition than the other). Based on the predictions, a PSC officer (PSCO) scheduling model is proposed to help the maritime authorities optimally allocate inspection resources. Considering that a PSCO can inspect at most four ships in a day, we further propose and incorporate the concepts of inspection template and un-dominated inspection template in the optimization models to reduce problem size as well as improve computation efficiency and model flexibility. Numerical experiments show that the proposed PSCO scheduling model with the predictions of XGBoost as the input is more than 20% better than the current inspection scheme at ports regarding the number of deficiencies detected. In addition, the gap between the proposed model and the model under perfect-forecast policy is only about 8% regarding the number of deficiencies detected. Extensive sensitivity experiments show that the proposed PSCO scheduling model has stable performance and is always better than the current model adopted at ports.

Horizontal Integrated Framework for Mobile Crowdsensing

Mobile crowdsensing is a promising paradigm to leverage the power of people to collect large-scale spatially distributed data. This concept has been intensely studied to efficiently and securely complete sensing tasks at lower cost. The development of a unified platform designed to provide various types of sensing applications is among the major approaches to economical crowdsourcing. However, existing previous frameworks were not optimized for shared use among multiple organizers because they were largely vertically integrated systems. Security and user trust and confidence is also a significant issue a crowdsensing frameworks, given the potential security concerns. Therefore, in this study, we propose a network-side task allocation (NeSTA) framework to address the existing problems in mobile crowdsensing. The proposed framework enables the horizontal integration of sensing applications, in which mobile networks mediate communication among organizers and participants, significantly reducing the installation cost of individual applications. Privacy preservation is achieved by task distribution and allocation procedures, where the participants were obscured by organizers. The validity of the proposed NeSTA was confirmed by simulations with an analytical model using an open dataset. The results show that the proposed method exhibited computational efficiency over two orders of magnitude greater than the conventional approach. This advantage originates from the reduction of problem size by dividing the original problem into subproblems.

Learning to Solve Large-Scale Security-Constrained Unit Commitment Problems

Security-constrained unit commitment (SCUC) is a fundamental problem in power systems and electricity markets. In practical settings, SCUC is repeatedly solved via mixed-integer linear programming (MIP), sometimes multiple times per day, with only minor changes in input data. In this work, we propose a number of machine learning techniques to effectively extract information from previously solved instances in order to significantly improve the computational performance of MIP solvers when solving similar instances in the future. Based on statistical data, we predict redundant constraints in the formulation, good initial feasible solutions, and affine subspaces where the optimal solution is likely to lie, leading to a significant reduction in problem size. Computational results on a diverse set of realistic and large-scale instances show that using the proposed techniques, SCUC can be solved on average 4.3 times faster with optimality guarantees and 10.2 times faster without optimality guarantees, with no observed reduction in solution quality. Out-of-distribution experiments provide evidence that the method is somewhat robust against data-set shift.Summary of Contribution. The paper describes a novel computational method, based on a combination of mixed-integer linear programming (MILP) and machine learning (ML), to solve a challenging and fundamental optimization problem in the energy sector. The method advances the state-of-the-art, not only for this particular problem, but also, more generally, in solving discrete optimization problems via ML. We expect that the techniques presented can be readily used by practitioners in the energy sector and adapted, by researchers in other fields, to other challenging operations research problems that are solved routinely.

Decomposition of arrow type positive semidefinite matrices with application to topology optimization

Decomposition of large matrix inequalities for matrices with chordal sparsity graph has been recently used by Kojima et al. (Math Program 129(1):33–68, 2011) to reduce problem size of large scale semidefinite optimization (SDO) problems and thus increase efficiency of standard SDO software. A by-product of such a decomposition is the introduction of new dense small-size matrix variables. We will show that for arrow type matrices satisfying suitable assumptions, the additional matrix variables have rank one and can thus be replaced by vector variables of the same dimensions. This leads to significant improvement in efficiency of standard SDO software. We will apply this idea to the problem of topology optimization formulated as a large scale linear semidefinite optimization problem. Numerical examples will demonstrate tremendous speed-up in the solution of the decomposed problems, as compared to the original large scale problem. In our numerical example the decomposed problems exhibit linear growth in complexity, compared to the more than cubic growth in the original problem formulation. We will also give a connection of our approach to the standard theory of domain decomposition and show that the additional vector variables are outcomes of the corresponding discrete Steklov–Poincaré operators.

Nurse to patient assignment through an analogy with the bin packing problem: Case of a Tunisian hospital

We describe the nurse to patient assignment problem that supports the operational decisions in the Neonatal Intensive Care Unit of Hedi Chaker Hospital. It implies efficiently assigning patients to nurses in their suitable unit of work, taking into consideration during assignment a set of patients having a given acuity to a set of nurses having certain capacities. A binary linear integer programming problem is formulated where each patient can be treated by an only one nurse. Due to the combinatorial complexity of the problem, a heuristic approach is developed to achieve the procedure of assignment. The Max–Max assignment heuristic is based on the development of four mathematical propositions that serve for insuring the problem size reduction. The resolution of the problem via the heuristic method has shown its pertinence for exact resolution using real instances extracted from the unit. For our real case the heuristic provided flexible modelling and gave a feasible and a good nurse to patient assignment for every shift and in each zone of assignment.

Performance-based inventory classification methods for large-Scale multi-echelon replenishment systems

Inventory classification is a managerial method utilized to group items that share predetermined characteristics, with the intent of assigning group-specific controls and monitoring mechanisms to each established item group. In this paper, we develop a performance-based inventory classification (PBIC) method that finds a grouping solution for a multi-item, multi-echelon inventory system controlled by continuous review. We argue that instead of grouping items based on similarities in unit cost, demand rate, or leadtime, the most effective strategy is to group items based on the information contained in their control policy values and their performance-related parameter values. We introduce a new policy-driven approach for establishing the classification criteria used to group items. We also adopt a ranking method to control the multi-dimensionality of multi-echelon systems in order to determine a one-dimension score. To group items, we improve the Pareto-based (ABC) solution by developing a search-based partitioning solution, utilizing a novel aggregation process. Our findings indicate that the PBIC method significantly outperforms alternative classification methods. Also, the empirical results show that there is a negligible gap between the performance of the PBIC and the optimal (complete enumeration) grouping solution. Finally, we discuss our work in the context of managerial implications highlighting the use of classification for problem aggregation and size reduction, when managers need to perform efficient, yet extensive, and dependable what-if analyses related to inventory management.

MineReduce: An approach based on data mining for problem size reduction

Hybrid variations of metaheuristics that include data mining strategies have been utilized to solve a variety of combinatorial optimization problems, with superior and encouraging results. Previous hybrid strategies applied mined patterns to guide the construction of initial solutions, leading to more effective exploration of the solution space. Solving a combinatorial optimization problem is usually a hard task because its solution space grows exponentially with its size. Therefore, problem size reduction is also a useful strategy in this context, especially in the case of large-scale problems. In this paper, we build upon these ideas by presenting an approach named MineReduce, which uses mined patterns to perform problem size reduction. We present an application of MineReduce to improve a heuristic for the heterogeneous fleet vehicle routing problem. The results obtained in computational experiments show that this proposed heuristic demonstrates superior performance compared to the original heuristic and other state-of-the-art heuristics, achieving better solution costs with shorter run times.

Inventory grouping and sensitivity analysis in multi-echelon spare part provisioning systems

Existing research in the area of multi-echelon spare part models have not adequately addressed how to take advantage of the useful concept of inventory segmentation. To fill this gap, we define a new formulation of the multi-echelon repairable parts stocking model for the purpose of finding the best inventory grouping solution. For solving this mixed-integer and nonlinear formulation, we develop a heuristic optimization model based on a greedy approach, which uses the idea that the items having more similar stocking policies should tend to group together. Our findings show that the proposed model provides a near optimal solution, which significantly outperforms the alternative classification and clustering methods in the field. In addition, we highlight the managerial implication of segmentation for problem size reduction, when managers need to perform an extensive sensitivity analysis under time limitations.

Architecture‐aware routability‐driven placer for large‐scale mixed‐size designs

In this study, the authors have presented a simple but robust routability-driven placement for the mixed-size designs. The proposed technique is implemented through (i) look-ahead legalisation-based global placement, (ii) congestion removal and (iii) detailed placement stages. A balanced clustering technique has been proposed to group the circuit blocks into clusters based on the types of circuit blocks and their connectivity. A 0-1 integer programming-based global placement method is framed, which performs look-ahead legalisation. A new site information table concept is introduced to keep the information about each placement cell. Based on the divide conquer strategy, placement area is divided into a region to reduce problem size. A force-directed method has been conceived to select an appropriate region for global placement for the blocks of a cluster. A new congestion removal approach substitutes the legalisation stage to remove circuit block congestion and pin density in different regions. Finally, a gain-based strategy has been introduced for routability-driven detailed placement. The proposed technique is implemented and tested on ICCAD 2012 benchmark circuits. It achieves 1.92 and 0.13% improvements in terms of half perimeter wirelength and routing congestion w.r.t. recent placers.