Local Search Improvement Research Articles

A mathematical model and solution algorithms for optimising system-level configurations of reconfigurable single part flow lines

This study addresses a system-level configuration selection problem for reconfigurable single part flow lines (R-SPFLs) with parallel identical flexible machines at each stage. The problem is to determine the number of stages, the number of machines of a certain type at each stage and the assignment of operations to each stage in order to satisfy the demand of a part type. Precedence relations among the operations and space limitations are also considered. For the objective of minimising the sum of machine purchase and operation processing costs, a nonlinear integer programming model is developed that gives the optimal solutions. Then, due to the problem's complexity, a basic variable neighbourhood search algorithm is proposed that constructs an initial solution using a greedy-type heuristic and improves it using a shaking and a local search improvement methods. Moreover, the basic algorithm is extended to a general one that uses a variable neighbourhood descent method. Computational results show that the two algorithms outperform the previous one significantly in both solution quality and computation times, and the general algorithm improves the basic one significantly. Finally, the applicability of the variable neighbourhood search algorithms is shown by reporting a brief case study.

A Hybrid Exact–Local Search Approach for One-Machine Scheduling with Time-Dependent Capacity

Machine scheduling is a hard combinatorial problem having many manifestations in real life. Due to the schedule followed, the possibility of installations of machines operating sub-optimally is high. In this work, we examine the problem of a single machine with time-dependent capacity that performs jobs of deterministic durations, while for each job, its due time is known in advance. The objective is to minimize the aggregated tardiness in all tasks. The problem was motivated by the need to schedule charging times of electric vehicles effectively. We formulate an integer programming model that clearly describes the problem and a constraint programming model capable of effectively solving it. Due to the usage of interval variables, global constraints, a powerful constraint programming solver, and a heuristic we have identified, which we call the “due times rule”, the constraint programming model can reach excellent solutions. Furthermore, we employ a hybrid approach that exploits three local search improvement procedures in a schema where the constraint programming part of the solver plays a central role. These improvement procedures exhaustively enumerate portions of the search space by exchanging consecutive jobs with a single job of the same duration, moving cost-incurring jobs to earlier times in a consecutive sequence of jobs or even exploiting periods where capacity is not fully utilized to rearrange jobs. On the other hand, subproblems are given to the exact constraint programming solver, allowing freedom of movement only to certain parts of the schedule, either in vertical ribbons of the time axis or in groups of consecutive sequences of jobs. Experiments on publicly available data show that our approach is highly competitive and achieves the new best results in many problem instances.

Scheduling algorithms for multi-stage flow shops with reworks under overlapped queue time limits

This study addresses a multi-stage flow shop scheduling problem in which a job is reworked when its queue time between two arbitrary stages exceeds an upper limit. The problem is to determine the start times of jobs and rework setups/operations if incurred for the objective of minimising makespan. As an extension of the previous studies, multiple overlapped queue time limits are considered, i.e. some in-between stages of queue time limits for a job are overlapped. A mixed integer programming model is developed and its performance is reported for small-sized test instances. Then, due to the limited applications of optimal solution approaches, a variable neighbourhood search (VNS) algorithm is proposed that generates an initial solution and improves it using a shaking and a local search improvement methods. In addition, it is extended to the general variable neighbourhood search (GVNS) algorithms with variable neighbourhood descent (VND) methods. Computational results show that the GVNS algorithms outperform the VNS algorithm significantly and also give near optimal solutions for small-sized test instances within a reasonable amount of computation times.

Delayed improvement local search

Local search is a fundamental tool in the development of heuristic algorithms. A neighborhood operator takes a current solution and returns a set of similar solutions, denoted as neighbors. In best improvement local search, the best of the neighboring solutions replaces the current solution in each iteration. On the other hand, in first improvement local search, the neighborhood is only explored until any improving solution is found, which then replaces the current solution. In this work we propose a new strategy for local search that attempts to avoid low-quality local optima by selecting in each iteration the improving neighbor that has the fewest possible attributes in common with local optima. To this end, it uses inequalities previously used as optimality cuts in the context of integer linear programming. The novel method, referred to as delayed improvement local search, is implemented and evaluated using the travelling salesman problem with the 2-opt neighborhood and the max-cut problem with the 1-flip neighborhood as test cases. Computational results show that the new strategy, while slower, obtains better local optima compared to the traditional local search strategies. The comparison is favourable to the new strategy in experiments with fixed computation time or with a fixed target.

The Traveling Salesman Problem with Job-times (TSPJ)

This paper explores a problem related to both the Traveling Salesman Problem and Scheduling Problem where a traveler moves through n locations (nodes), visits all locations and each location exactly once to assign and initiate one of n jobs, and then returns to the first location. After initiation of a job, the traveler moves to the next location immediately and the job continues autonomously. This is representative of many practical scenarios including autonomous robotics, equipment maintenance, highly automated manufacturing, agricultural harvesting, and disaster recovery. The goal is to minimize the time of completion of the last job, i.e. makespan. Although the makespan objective typically is used in scheduling problems, the transportation times can be significant relative to the processing time, and this changes the optimization considerations. We refer to this problem as the Traveling Salesman Problem with Job-times (TSPJ) and present a mathematical formulation and local search improvement heuristics for it. Computational experience in solving the TSPJ using both commercial solvers and heuristics is reported.

A variable neighborhood search algorithm with reinforcement learning for a real-life periodic vehicle routing problem with time windows and open routes

This paper studies a real-life container transportation problem with a wide planning horizon divided into multiple shifts. The trucks in this problem do not return to depot after every single shift but at the end of every two shifts. The mathematical model of the problem is first established, but it is unrealistic to solve this large scale problem with exact search methods. Thus, a Variable Neighbourhood Search algorithm with Reinforcement Learning (VNS-RLS) is thus developed. An urgency level-based insertion heuristic is proposed to construct the initial solution. Reinforcement learning is then used to guide the search in the local search improvement phase. Our study shows that the Sampling scheme in single solution-based algorithms does not significantly improve the solution quality but can greatly reduce the rate of infeasible solutions explored during the search. Compared to the exact search and the state-of-the-art algorithms, the proposed VNS-RLS produces promising results.

A hybrid genetic and Lagrangian relaxation algorithm for resource-constrained project scheduling under nonrenewable resources

Scheduling under nonrenewable resources is one of the challenging issues in project scheduling problems. There are many cases where the projects are subject to some nonrenewable resources. In most of the literature, nonrenewable resources are assumed to be available in full amount at the beginning of the project. However, in practice, it is prevalent that these resources are procured along the project horizon. This paper studies the generalized resource-constrained project scheduling problem (RCPSP) where, in addition to renewable resources, nonrenewable resources are considered, such as budget or consuming materials by the project activities. As the problem is NP-hard, some sub-algorithm elements are developed, which can be used in the structure of inexact approaches for solving the problem. These elements include constraint propagation, priority rules, schedule generation schemes, and local search improvement procedures. Also, a lower bounding algorithm is developed based on the Lagrangian Relaxation (LR) approach, and the problem is optimized by the Genetic Algorithm (GA). The hybrid GA–LR​ algorithm produces a result reasonably near to optimum solutions. Comprehensive computational experiments based on standard project scheduling problems are performed to evaluate these developments. The experiments showed the performance and robustness of the proposed algorithm.

Metaheuristic Algorithms for the Many-to-one IRP with Dynamic Velocity

Inventory Routing Problem (IRP) is a combination of inventory management and transportation optimization problems that involve route selection, number of product pickups, and customer demands. In the many-to-one IRP model, the vehicle is sent from the depot and goes to pick up products from several suppliers to the assembly plant. Vehicle loads on this model will increase every time a product is taken from the supplier. It can decrease vehicle speed in the next travel process (dynamic velocity). There are still a few papers that discuss many-toone IRP models with dynamic velocity. Therefore, this study aims to develop a new model called the many-to-one IRP model with dynamic velocity. A modified threshold accepting, variable neighborhood search, and record-to-record travel algorithm with the first improvement local search strategy is used to solve the IRP model. Small datasets many-to-one IRP from previous researches and experimental tests are used to test the algorithms. The results showed that the best-proposed algorithm is competitive when compared to the best-known solution in the previous studies (the average deviation is only 1.86%).

A hybrid meta-heuristic algorithm for scientific workflow scheduling in heterogeneous distributed computing systems

Cloud computing has attracted great attentions in research community because of its ubiquitous, unlimited computing resources, low cost, and flexibility owing to virtualization technology. This paper presents a hybrid meta-heuristic algorithm to solve parallelizable scientific workflows on elastic cloud platforms since applying a single approach cannot yield optimal solution in such complicated problems. Scientific workflows are modeled in the form of directed acyclic graph (DAG) in which there exists data dependency between sub-tasks. In the cloud marketplace, each provider delivers variable virtual machine (VM) configurations which lead different performance. Generally, parallelizable task scheduling on parallel computing machines to obtain minimum total execution time, makespan, belongs to NP-Hard problem. To deal with the combinatorial problem, the hybrid discrete particle swarm optimization (HDPSO) algorithm is presented that has three main phases. At the first phase a random algorithm following by novel theorems is applied to produce swarm members; it is as input of presented new discrete particle swarm optimization (DPSO) algorithm in the second phase. To avoid getting stuck in sub-optimal trap and to balance between exploration and exploitation, local search improvement is randomly combined in DPSO by calling Hill Climbing technique at the third phase to enhance overall performance. Second and third phases are iterated till the termination criteria is met. The average results reported from different executions of intensive settings on 12 scientific datasets proved our hybrid meta-heuristic has the amount of 10.67, 14.48, and 3 percentage dominance in terms of SLR, SpeedUp, and efficiency respectively against other existing meta-heuristics.

Metaheuristic Approaches for One-Dimensional Bin Packing Problem: A Comparative Performance Study

Nature-inspired metaheuristic algorithms have steadily gained popularity over the last two decades. They have been applied to a plethora of optimization problems both in continuous and combinatorial domains. In this paper, the one-dimensional bin packing problem is solved through the implementation of two underlying heuristics, namely, best fit and better fit, and four representative state-of-the-art global metaheuristic algorithms, namely, firefly algorithm, genetic algorithm, adaptive cuckoo search algorithm, and artificial bee colony algorithm. The underlying best fit and better fit heuristics are employed by the four aforementioned global metaheuristic algorithms as reordering heuristics and local search improvement mechanisms are used for generating good packing schema. Furthermore, these two local heuristics possess special characteristics which, when incorporated into the global metaheuristics, allow them to escape local optimums and avoid getting stuck, and more so, enables the algorithms to generate good quality solutions. The main focus of this paper is the presentation of a systematic performance evaluation study for the representative algorithms, with some initial computational results that show the effectiveness of the respective algorithms and their ability to achieve promising solutions. The experiments conducted here were carried out using three standard bin packing problem datasets categories with over 1,210 instances in total, each with differing capacities, number of items and distributions between item weights. The numerical results of the representative algorithms were compared with the solutions achieved with the underlying heuristic techniques, in this case, the best fit and better fit heuristics, respectively. Similarly, the analysis of the initial computational results obtained revealed the superior performance of the individual algorithm implementation. Moreover, performance was established by taking into account both the algorithms computational time and the solution quality. Overall, several observations regarding the solution and the underlying heuristics were made. It is worth noting that by utilizing best fit heuristic, the algorithms attained optimal solutions for instances of the easy dataset requiring smaller capacity bins. However, as the complexity of the instances increased, the ability to produce high-quality results decreased. Nevertheless, this heuristic can produce near-optimal solutions and a good packing schema for most instances. On the other hand, utilizing better fit as the underlying heuristic results in optimal solutions almost all the time, regardless of capacity and number of items.

Managing appointments with waiting time targets and random walk-ins

The random arrivals of walk-in patients significantly affect the daily operations of healthcare facilities. To improve the performance of outpatient departments, this paper attempts to make an appointment schedule by considering walk-ins and the waiting time target (WTT) for appointment patients. A stochastic programming model is proposed to solve this problem with the objective of minimizing the weighted patient waiting and makespan cost. A non-decreasing waiting cost function is used to capture the WTT fulfillment of appointment patients, whereas walk-ins incur a linear waiting cost. A finite-horizon Markov Decision Process model is formulated to establish the optimal real-time scheduling policy under a given appointment schedule. The appointment schedule is determined by a two-stage stochastic programming approximation and a local search improvement. Structural properties of the optimal appointment scheduling and real-time scheduling policies are established. In particular, it is shown that appointment overbooking is allowed only at the end of the regular session, and the optimal real-time scheduling policy is an easy-to-implement threshold policy with bounded sensitivity. Numerical experiments based on real data are performed to investigate the influence of different parameters and to compare different schedules. The optimal schedule demonstrates superior performance by allowing reasonable waiting times for appointment patients depending on their WTTs. Managerial insights are also provided to hospital managers. Finally, the basic model is extended by incorporating random service times and random arrivals of appointment patients. The latter includes the random number of patients that show up for service or call for appointments, and the random arrival time (unpunctuality). Appointment overbooking strategies are shown to have different structures under some stochastic factors.

Efficient local search strategies for the Mixed Capacitated Arc Routing Problems under Time Restrictions with Intermediate Facilities

In this paper, we extend our previous work on greedy constructive heuristics for the Mixed Capacitated Arc Routing Problem under Time Restrictions with Intermediate Facilities (MCARPTIF) by developing efficient local search improvement heuristics for the problem. Five commonly used arc routing move operators were adapted for the problem, and basic local search implementations were tested on waste collection benchmark sets. Tests showed that despite the application of commonly used speed-up techniques, the local search implementations are very slow on large test instances with more than one-thousand required arcs and edges. In response, more advanced local search acceleration mechanisms from literature were adapted and combined for the MCARPTIF and tested on the same instances. On the large instances, the basic local search setups took between 15 min and 3 h to improve a single solution to local optima, whereas the accelerated implementations took at most 4 min while producing similar quality local optima. The best performing implementation made use of two existing acceleration mechanisms, namely Static-Move-Descriptors and Greedy-Compound-Independent-Moves. A third mechanism, Nearest-Neighbour-Lists was also tested and although it reduced execution times it resulted in local search terminating at worse quality local optima. Given the importance of local search within metaheuristics, the developed local search heuristics provide a significant contribution to arc routing. First, the implementations can be extended to and incorporated into metaheuristics for the MCARPTIF. Second, the acceleration mechanisms can be applied to existing local search metaheuristics for Capacitated Arc Routing Problems, thereby improving the efficiency of the metaheuristics and allowing them to better deal with large instances.

Alternative evaluation functions for the cyclic bandwidth sum problem

One essential element for the successful application of metaheuristics is the evaluation function. It should be able to make fine distinctions among the potential solutions in order to avoid producing wide plateaus (valleys) in the fitness landscape, on which detecting a promising search direction could be hard for certain local search strategies. In the specific case of the cyclic bandwidth sum (CBS) problem, the heuristics reported have used directly the objective function of the optimization problem to assess the quality of potential solutions. Nevertheless, such a conventional function does not allow to efficiently establish preferences among distinct potential solutions. In order to cope with this important issue, three new more refined evaluation functions for the CBS problem are introduced in this paper.An in-depth comparative analysis considering the conventional and the three proposed evaluation functions is carried out and presented. It includes an assessment of their: (a) discrimination potential, (b) consistency with regard to the primary objective of the CBS problem, and (c) practical usefulness within two different algorithms, best improvement local search and iterated local search. A validation of the experimental results by means of a meticulous statistical significance analysis revealed that proposing more informative evaluation schemes for the CBS problem could be a useful means of improving the performance of metaheuristics. Indeed, our iterated local search implementation, using an alternative evaluation function, surpassed the best solutions yielded by the state-of-the-art algorithms and allow us to attain new better upper bounds for 14 out of 20 well-known benchmark instances.

Effective heuristic for large-scale unrelated parallel machines scheduling problems

This paper is concerned with non-preemptive scheduling of large-scale unrelated parallel machines (UPM) with the objective of minimizing total weighted completion times (TWCT). We propose a sequential improvement local search algorithm using multiple-jump strategy embedded within Tabu search (TS) components for TWCT, and use a highly efficient data structure to provide a necessary and sufficient condition for local optimality of a solution. We will generate a set of large-scale test problems to evaluate the performance of proposed algorithm in term of scalability, solution quality and efficiency. The non-parametric tests of algorithm components will be used to validate the consistent performance across problem types and problem sizes in the proposed algorithm.

General variable neighborhood search for solving Sudoku puzzles: unfiltered and filtered models

In this paper, two novel models based on the Variable Neighborhood Search (VNS) algorithm are proposed to solve Sudoku puzzles. For the first model (Unfiltered-VNS), four neighborhood structures are proposed. The Filtered-VNS, which is the second model, uses filtering to reduce the number of partial infeasible solutions in the search area. Local search is performed by a novel mutation-based neighborhood structure. In both models, the neighborhood structures are implemented by using different local search improvement strategies. Two proposed models with the best configurations are tested on 57 well-known Sudoku benchmarks. The experimental results indicate that our models can solve all benchmarks. For easy-, medium-, and hard-level puzzles, Filtered-VNS shows better solution quality than Unfiltered-VNS. For very hard instances, performance of Unfiltered-VNS is better than Filtered-VNS. Except for two of the 57 benchmarks, Filtered-VNS improves the solution qualities of the previous studies.

Heuristics for the weighted k-rural postman problem with applications to urban snow removal

We describe a weighted version of the k-Chinese and k-rural postman problem that occurs in the context of snow removal. The problem concerns the questions of which vehicle shall take care of each link and how the vehicles shall travel between links. We also consider different numbers of vehicles, in view of a fixed cost for each vehicle. We describe and discuss heuristic solution approaches, based on usable substructures, such as heuristics for rural postman problems, meta-heuristics, k-means clustering and local search improvements by moving cycles. The methods have been implemented and tested on real life examples.

Relaxations and heuristics for the multiple non-linear separable knapsack problem

We consider the multiple non-linear knapsack problem with separable non-convex functions. The problem, which can be modeled as a (mixed) integer non-linear program, is extremely difficult to solve in practice. We present a fast heuristic algorithm, based on constructive techniques, surrogate relaxations, and local search improvements. Computational comparisons with exact and heuristic methods for general non-convex mixed integer non-linear programs show that the proposed approach provides good-quality solutions within small computing times.

Simple and fast novel diversification approach for the UBQP based on sequential improvement local search

The unconstrained binary quadratic program (UBQP) is a challenging NP-hard problem. Due to its vast applicability and the theoretical interests it has grown in importance in the recent years. Various heuristics have been proposed as solution procedures. Most of the heuristics are based on local improvement procedures. To be able to reach optimal or near optimal solutions, researchers have implemented multistart and diversification strategies to explore a larger solution space. Diversification strategies in the literature concentrate on some manipulation of solutions (variables). In this paper, relationship between starting solution, the sequence of implementing local search, and the locally optimal solution x∗ is explored. A novel diversification approach based on the sequence to implement the local improvement is proposed. We implemented four versions of our diversification procedures within a simple tabu search and tested on several benchmark problems available on the Internet. Our extensive computational results show that the procedures can reach the best known solutions with high frequencies within very short CPU time. For 123 out of 125 of these problems the procedures reached the best known solutions quickly. For 44 of the 84 Max-Cut benchmark problems the procedures improved upon the available solutions in reasonably short CPU time.

A hybrid binary particle swarm optimization for large capacitated multi item multi level lot sizing (CMIMLLS) problem

The lot sizing problem deals with finding optimal order quantities which minimizes the ordering and holding cost of product mix. when multiple items at multiple levels with all capacity restrictions are considered, the lot sizing problem become NP hard. Many heuristics were developed in the past have inevitably failed due to size, computational complexity and time. However the authors were successful in the development of PSO based technique namely iterative improvement binary particles swarm technique to address very large capacitated multi-item multi level lot sizing (CMIMLLS) problem. First binary particle Swarm Optimization algorithm is used to find a solution in a reasonable time and iterative improvement local search mechanism is employed to improvise the solution obtained by BPSO algorithm. This hybrid mechanism of using local search on the global solution is found to improve the quality of solutions with respect to time thus IIBPSO method is found best and show excellent results.

An ensemble of intelligent water drop algorithms and its application to optimization problems

The Intelligent Water Drop (IWD) algorithm is a recent stochastic swarm-based method that is useful for solving combinatorial and function optimization problems. In this paper, we propose an IWD ensemble known as the Master-River, Multiple-Creek IWD (MRMC-IWD) model, which serves as an extension of the modified IWD algorithm. The MRMC-IWD model aims to improve the exploration capability of the modified IWD algorithm. It comprises a master river which cooperates with multiple independent creeks to undertake optimization problems based on the divide-and-conquer strategy. A technique to decompose the original problem into a number of sub-problems is first devised. Each sub-problem is then assigned to a creek, while the overall solution is handled by the master river. To empower the exploitation capability, a hybrid MRMC-IWD model is introduced. It integrates the iterative improvement local search method with the MRMC-IWD model to allow a local search to be conducted, therefore enhancing the quality of solutions provided by the master river. To evaluate the effectiveness of the proposed models, a series of experiments pertaining to two combinatorial problems, i.e., the travelling salesman problem (TSP) and rough set feature subset selection (RSFS), are conducted. The results indicate that the MRMC-IWD model can satisfactorily solve optimization problems using the divide-and-conquer strategy. By incorporating a local search method, the resulting hybrid MRMC-IWD model not only is able to balance exploration and exploitation, but also to enable convergence towards the optimal solutions, by employing a local search method. In all seven selected TSPLIB problems, the hybrid MRMC-IWD model achieves good results, with an average deviation of 0.021% from the best known optimal tour lengths. Compared with other state-of-the-art methods, the hybrid MRMC-IWD model produces the best results (i.e. the shortest and uniform reducts of 20 runs) for all13 selected RSFS problems.