Nested Partitions Algorithm Research Articles

Efficient heuristic methods for berth allocation at multi-line, multi-berth curbside bus stops

Transit management agencies often pre-allocate a multi-berth stop’s berths to specific bus lines so that passengers can find the right place to wait for their buses. Developing optimal berth allocation plans that minimize the total bus delay is essential for mitigating bus queues at busy multi-berth stops. However, this problem is challenging due to the huge solution space, the high degree of stochasticity in bus queues, and the resulting extremely high computational cost. In this paper, we first propose a simple heuristic method inspired by queueing theory. It is based on the idea that evenly distributing the total traffic intensity (defined as the total bus arrival rate times the mean dwell time) among all the berths would produce a lower bus delay. Numerical results demonstrate that this simple method generated very good berth allocation plans (with optimality gaps <6% for no-overtaking and free-overtaking stops) in seconds! It does not rely on time-consuming simulation surrogate models or numerous input data such as the stochastic bus arrival processes and dwell time distributions of each bus line. To further improve the simple heuristic’s performance (especially for limited-overtaking stops), we develop a cluster-based nested partition algorithm that can find a near-optimal plan (e.g., with an optimality gap of <3%) in a much shorter time than a previous algorithm. The algorithm employs the simple heuristic plan as the initial solution. Our methods can be applied to stops with various berth numbers, different proximities to nearby traffic signals, and under diverse bus queueing rules.

A framework using nested partitions algorithm for convergence analysis of population distribution-based methods

Stochastic optimization algorithms such as genetic algorithm (GA), particle swarm optimization (PSO), estimation of distribution algorithms (EDAs), and nested partitions algorithm (NPA) are used in many problems including nonlinear model predictive control and task assignment. Some of these algorithms, however, lack global convergence guarantee such as PSO, or require strict convergence assumptions such as NPA. To enhance these methods in terms of convergence, a common underlying framework towards representing the seemingly unrelated methods is established as the updating of the distribution of the population through iterative sampling, and the methods that fit into this framework are called population distribution-based methods. Global convergence conditions for this framework are innovatively developed by building a shadow NPA structure for the population evolution process. The result is generic and is capable of analyzing convergence of many methods including GA, PSO, EDA, and NPA. It can be further exploited to improve convergence by modifying these methods. The existing and modified variants of these methods are then applied to case studies to show the improvement.

Optimal station locations for en-route charging of electric vehicles in congested intercity networks: A new problem formulation and exact and approximate partitioning algorithms

This paper addresses a new electricity-charging station location problem for congested intercity or regional networks, in which electric vehicles with a limited driving range require recharges for their long-distance trips. When the distribution of charging stations does not sufficiently cover all used routes, some drivers may take a detour to find charging opportunities (or switch to another transportation mode). The goal of this station location problem is to find, under a limited construction budget, an optimal set of charging station locations such that all vehicles finish their trips by choosing a self-optimal route with necessary charging opportunities while the overall network congestion caused by possible detours is minimized. The problem is written as a bi-level mixed nonlinear integer programming model, where the upper level of the model is set to regulate the selection of station locations subject to the construction budget while the lower level is used to characterize the equilibrium flow pattern of electric vehicles with the charging requirement. Selected exact and approximate algorithms, namely, the branch-and-bound algorithm and the nested partitions algorithm are adopted to solve this station location problem. While both algorithms imply a divide-and-conquer strategy, the branch-and-bound algorithm poses a deterministic, exact procedure that utilizes the bounding mechanism to prune impossible solution subspaces, whereas the nested partitions algorithm performs a stochastic search and selection process in terms of the optimality probability for near-optimal solutions. To get numerical insights on the algorithmic performance and solution behavior, we test these algorithms through a couple of benchmark network instances. A performance comparison of the algorithms indicates that, the branch-and-bound algorithm can quickly obtain the global optimum when the driving range is relatively low, while the nested partitions algorithm can find optimal solutions or extremely near-optimal solutions in all cases and typically spend on average only one fourth of the computing time of the branch-and-bound algorithm. The network flow solutions clearly show, compared to gasoline vehicle drivers, how an insufficient driving range or number of charging stations may significantly reduce the number of feasible paths and force electric vehicle drivers to choose more costly paths, which thus reshapes flow patterns and possibly increases congestion levels.

Using Contingent Valuation and Numerical Methods to Determine Optimal Locations for Environmental Facilities: Public Arboretums in South Korea

Because the social benefit generated from a public facility varies according to its accessibility, population density of location, local income level, local taste, and the existence of similar nearby facilities, the value of facilities can vary depending on their location. Therefore, determining the ideal location is an important problem. This study provides a new direction for solving this problem by combining the contingent valuation (CV) method and nested partitions (NP) algorithm using siting of public arboretums as an example. First, consumers' willingness to pay for environmental facilities considering household characteristics and distance to both newly built and existing facilities are estimated using the CV method. In this process, consumers are divided into two groups (active and passive users), assuming they have different willingness to pay for the facilities. Then, using the results of the CV, the NP algorithm, a simulation-based discrete optimization technique, is constructed to efficiently identify optimal locations that maximize social benefit considering regional characteristics. The results of the proposed algorithm exceeded the performance of the benchmark case, and this study's findings can be used to aid decisions about complicated multiple facility locations. Moreover, visualized results are provided, which can be useful for local and central decision makers.

Minimizing Completion Time for Order Scheduling: Formulation and Heuristic Algorithm

In this study, the customer order scheduling problem is investigated to minimize total weighted completion time. A quadratic formulation is proposed to address this problem. This formulation is converted into an equivalent mixed-integer linear programming model by applying the linearization technique and the special structure of the problem. The problem size that can be solved to optimality is then investigated and reported based on the final linearized formulation. Furthermore, a hybrid nested partitions algorithm is developed to solve large-scale problems. Numerical results illustrate the advantages of the proposed model and demonstrate that the proposed algorithm can obtain high-quality solutions within a reasonable computational time.

An activity-list-based nested partitions algorithm for resource-constrained project scheduling

The resource-constrained project scheduling problem (RCPSP) has received wide attention. In this paper, an activity-list-based nested partitions algorithm (ALNP) is developed for solving the RCPSP and a P-ALNP is proposed to improve ALNP with local adjustment. In the algorithms, to improve the search efficiency, a partial double justification is employed as local search mechanism. The computational experiments on the PSPLIB and analysis on robustness of the algorithms show that ALNP outperforms the traditional serial scheduling scheme for solving the large-scale, complex RCPSPs, and P-ALNP can improve ALNP and obtain better results. P-ALNP is a competitive algorithm for solving the RCPSP.

An energy-responsive optimization method for machine tool selection and operation sequence in flexible machining job shops

The environmental burden caused by energy consumption during the use phase of machine tool systems is widely acknowledged and hence ways must be found to use energy more efficiently. There is potentially a significant amount of energy savings that could be realized by selecting alternative machine tools and reducing the idle energy consumption through better scheduling. This paper proposes an energy-saving optimization method that considers machine tool selection and operation sequence for flexible job shops. The former seeks to reduce the energy consumption for machining operations, and the latter aims to reduce the idle energy consumption of machine tools. A mathematical model is formulated using mixed integer programming and the energy consumption objective is combined with a classical objective, the makespan. A Nested Partitions algorithm, which has proved to be robust for NP-hard problems, is utilized to solve the model. The proposed method is evaluated in a test case by two scenarios with different energy optimization schemes as well as the classical makespan objective. The results show that the proposed method is effective at realizing energy-savings for a flexible machining job shop.

Nested partitions for global optimization in nonlinear model predictive control

In Nonlinear Model Predictive Control (NMPC), the optimization problem may be nonconvex. It is important to find a global solution since a local solution may not be able to operate the process at desired setpoints. Also the solution must be available before the control input has to be applied to the process. In this paper, a stochastic algorithm called the Nested Partitions Algorithm (NPA) is used for global optimization. The NPA divides the search space into smaller regions and either concentrates search in one of these regions called the most promising region or backtracks to a larger region in the search space based on a performance index. To adapt the NPA to solve dynamic NMPC with continuous variables, a new partitioning scheme is developed that focuses on the first few control moves in the control horizon. The expected number of iterations taken by the NPA is presented. Convergence speed is improved by reducing the size of the starting most promising region based on a good starting point. The discrete sampling nature of the NPA may cause difficulty in finding the global solution in a continuous space. A gradient-based search is used with the NPA to overcome this difficulty. The solution quality is assessed in terms of the error from the actual global minimum. The algorithm is shown to give a feasible solution that provides asymptotic stability. Case studies are used to show the algorithm performance in terms of tracking setpoints, cost, solution quality and convergence time.

A hybrid nested partitions algorithm for scheduling flexible resource in flow shop problem

Production scheduling with flexible resources is critical and challenging in many modern manufacturing firms. This paper applies the nested partitions (NP) framework to solve the flexible resource flow shop scheduling (FRFS) problem using an efficient hybrid NP algorithm. By considering the domain knowledge, the ordinal optimisation principle and the NEH heuristics are integrated into the partitioning scheme to search the feasible region. An efficient resource-allocation procedure is built into the sampling scheme for the FRFS problem. A large number of benchmark examples with flexible resources are tested. The test results show that the hybrid NP algorithm is more efficient than either generic NP or heuristics alone. The algorithm developed in this study is capable of selecting the most promising region for a manufacturing system with a high degree of accuracy. The algorithm is efficient and scalable for large-scale problems.

Weighted nested partitions based on differential evolution (WNPDE) algorithm-based scheduling of parallel batching processing machines (BPM) with incompatible families and dynamic lot arrival

The scheduling problem of parallel batching processing machines (BPM) with incompatible families and dynamic lot arrivals in the diffusion and oxidation of semiconductor production line is studied. Given that the batch scheduling problem is NP-hard, it is decomposed into three sub-problems: batch forming, machine assignment and batch sequencing. A novel algorithm named weighted nested partitions based on differential evolution (WNPDE) is proposed for the most critical sub-problem: machine assignment sub-problem. The WNPDE retains the global perspective of nested partitions algorithm and the local search capabilities of differential evolution (DE) algorithm. Experimental results indicate that the proposed WNPDE algorithm has high efficiency to minimise the total weighted tardiness (TWT).

An Enhanced Nested Partitions Algorithm Using Solution Value Prediction

Metaheuristics are an important branch of optimization algorithms that attract lots of research and application efforts. In this paper, the research of predicting solution value for Nested Partitions (NP) is proposed, which is a newly developed metaheuristic algorithm for solving large-scale optimization problems. The lower bound embedded prediction procedures are developed to predict the future performance of NP based on the solution values obtained at early iterations. The prediction procedures are used in an enhanced NP algorithm to select a proper algorithm setting for NP at early stage, which saves a lot of computational resource for large-scale problems. The computational tests show the accuracy and effectiveness of the proposed algorithm. These prediction procedures can be also applied to some other metaheuristics.

A Hybrid Nested Partitions Algorithm Combing with Genetic Algorithm for Job Shop Flexible Resource Scheduling

In this paper, we formulate the Job Shop Flexible Resource Scheduling (JSFRS) problem, and presents a hybrid nested partitions algorithm combining genetic algorithm in sampling scheme for the JSFRS problem. The major part of this work is a generic partitioning scheme was implemented and the Resource Allocation Procedure was incorporated into the NP method framework. Simulation results support our theoretical results and illustrate that the hybrid nested partitions algorithm work as designed and the performance improvements associated with flexible resource scheduling are substantial.

A Hybrid Nested Partitions Algorithm for Banking Facility Location Problems

The facility location problem has been studied in many industries including banking network, chain stores, and wireless network. Maximal covering location problem (MCLP) is a general model for this type of problems. Motivated by a real-world banking facility optimization project, we propose an enhanced MCLP model which captures the important features of this practical problem, namely, varied costs and revenues, multitype facilities, and flexible coverage functions. To solve this practical problem, we apply an existing hybrid nested partitions algorithm to the large-scale situation. We further use heuristic-based extensions to generate feasible solutions more efficiently. In addition, the upper bound of this problem is introduced to study the quality of solutions. Numerical results demonstrate the effectiveness and efficiency of our approach.

A hybrid of Nested Partition, Binary Ant System, and Linear Programming for the multidimensional knapsack problem

This work presents a hybrid algorithm of Nested Partition (NP), Binary Ant System (BAS), and Linear Programming (LP) to solve the multidimensional knapsack problem (MKP). The hybrid NP+BAS+LP algorithm takes advantage of the global search strategies of the NP algorithm; the ability of BAS to quickly generate good solutions and incorporates information obtained from solving a LP relaxation of the MKP to help guide the search. It thus incorporates both the lower bounds (LB), found by the BAS, and the upper bounds (UB), found by solving the relaxed LP, into the search by embedding both in the NP framework. An adjustable computation budget is implemented where the number of samples increases if the LB and the UB point to different promising subregions. The proposed hybrid is compared to state-of-the-art solution techniques and is found to be one of the best algorithms in terms of the quality of solutions obtained and CPU time requirements.

A new hybrid algorithm for feature selection and its application to customer recognition

An important problem in customer recognition process is to select the most valid customer features. The authors present a new optimisation algorithm that combines a global optimisation algorithm called the nested partitions algorithm and the simulated annealing method. The resulting hybrid algorithm NP/SA retains the global perspective of the nested partitions algorithm and the local search capabilities of the simulated annealing method. A detailed application of the new algorithm to a customer recognition problem is also presented. The numerical results suggest that the new framework has great computation efficiency and convergence speed and is very efficient for a difficult customer feature selection problem.

An improved nested partitions algorithm based on simulated annealing in complex decision problem optimization

This paper introduces the main ideas of the nested partitions (NP) method, analyses its efficiency theoretically and proposes the way to improve the optimization efficiency of the algorithm. Then t...

SU‐FF‐T‐10: A Distributed‐Computing Framework Using Nested Partitions for Beam‐Angle Optimization in IMRT

Purpose: Commercial IMRT treatment planning software currently solves the dose optimization (DO) problem, but does not consider additional variables, such as gantry, collimator, or couch angles. These parameters are generally specified by clinicians based on experience, and are often arbitrary and suboptimal. Including the additional variables in the treatment planning process increases the size of the combinatorial problem, and would require access to vast computational resources and novel optimization techniques. We have formed a collaboration with a leading academic supercomputer center and implemented the Nested Partitions (NP) algorithm, a versatile and efficient method, to produce IMRT plans which employ optimal beam angles. Method and Materials: NP is a metaheuristic algorithm which is flexible enough to guide the search of a heuristic or deterministic dose optimization algorithm, and is well suited for implementation on distributed‐computing platforms. We implemented the NP framework in conjunction with the dose‐optimization algorithm employed in a commercial treatment planning system. We performed a study on a 5‐field prostate case by executing the programs on four machines simultaneously, and evaluated the resulting treatment plans based on DVH constraints for the target and critical structures. Results: After only three iterations of the NP algorithm, the resulting plan realized an improvement of 14% in a measure of beam‐angle‐sample quality over a clinical plan employing equi‐spaced beams. Since each iteration was implemented simultaneously on multiple machines, a decrease in execution time by a factor of about 4 was observed. Conclusion: Including additional variables, such as beam angles, in the IMRT optimization problem results in a vastly increased search space. However, our results indicate that the use of a distributed‐computing platform combined with a powerful metaheuristic optimization algorithm can result in treatment plans which provide better target coverage and normal tissue sparing, and can be generated more efficiently than those currently produced.

SU-FF-T-30: A Nested Partitions Framework for Beam Angle and Dose Optimization in IMRT

Purpose: To present a novel algorithm, nested partitions (NP), capable of finding suitable beam angle samples for IMRTtreatment planning by guiding the dose optimization process. Beam angle optimization and dose optimization are two problems which are conventionally solved separately, because coupling the variables increases the size and complexity of the combinatorial optimization problem. Method and Materials: NP is a metaheuristic algorithm, guiding the search of a deterministic dose optimization algorithm. The NP method adaptively samples from the entire feasible region, or search space, and concentrates the sampling effort by systematically partitioning the feasible region at successive iterations. We used a “warm‐start” approach by initiating the NP with beam angle samples derived from an integer programming (IP) model. We implemented the NP framework in conjunction with a quasi‐newton dose‐optimization algorithm employed in a commercial treatment planning system. We evaluated the results using 7‐field plans for two test clinical cases: head and neck and pancreas. Results: The results of four iterations of the NP algorithm outperformed both the initial IP solution and a generic equi‐spaced beam angle plan. This evaluation was based on DVH constraints for the critical structures for both clinical cases. For example, in the head and neck case, the NP plan delivered a dose of greater than 35 Gy to just 4.3% of the spinal cord, compared to 5.2% for the IP plan and 41.4% for the generic plan. In the pancreas case, the NP delivered a dose of greater than 23 Gy to 30.9% of the right kidney, compared to 43.7% (IP plan) and 49.0% (generic plan). Conclusions: Our results indicate that the IP solution provides a good initial solution. In addition, by employing the NP framework, further improvement is achieved. This makes it possible to produce a high‐quality solution within a reasonable amount of time.