Parallel variable neighbourhood search algorithms for job shop scheduling problems

^** Email: msevkli{at}fatih.edu.tr^*** Corresponding author. Email: mehmetaydin{at}acm.org, mehmet.aydin{at}beds.ac.uk Variable neighbourhood search (VNS) is one of the most recentmetaheuristics used for solving combinatorial optimization problemsin which a systematic change of neighbourhood with a local searchis carried out. However, as happens with other metaheuristics,it takes a long time to reach some useful solutions while solvingsome sort of hard combinatorial problems such as job shop scheduling(JSS). Parallelization is one of the most considerable policiesto overcome this matter. In this paper, firstly, a number ofVNS algorithms are examined for JSS problems and then four differentparallelization policies are taken into account to determineefficient parallelization for VNS algorithms. The experimentationreveals the performance of various VNS algorithms and the efficiencyof policies to follow in parallelization. In the end, the unilateral-ringtopology, a noncentral parallelization method, is found as themost efficient policy.     

Scheduling jobs on parallel machines with sequence-dependent setup times

Consider a number of jobs to be processed on a number of identical machines in parallel. A job has a processing time, a weight and a due date. If a job is followed by another job, a setup time independent of the machine is incurred. A three phase heuristic is presented for minimizing the sum of the weighted tardinesses. In the first phase, as a pre-processing procedure, factors or statistics which characterize an instance are computed. The second phase consists of constructing a sequence by a dispatching rule which is controlled through parameters determined by the factors. In the third phase, as a post-processing procedure, a simulated annealing method is applied starting from a seed solution which is the result of the second phase. In the dispatching rule of the second phase there are two parameters of which the values are dependent on the particular problem instance at hand. Through extensive experiments rules are developed for determining the values of the two parameters which make the priority rule work effectively. The performance of the simulated annealing procedure in the third phase is evaluated for various values of the factors.     

Tabu search for multi-mode resource-constrained project scheduling with schedule-dependent setup times

In this paper, a multi-mode resource-constrained project scheduling problem with schedule-dependent setup times is considered. A schedule-dependent setup time is defined as a setup time dependent on the assignment of resources to activities over time, when resources are, e.g., placed in different locations. In such a case, the time necessary to prepare the required resource for processing an activity depends not only on the sequence of activities but, more generally, on the locations in which successive activities are executed. Activities are non-preemptable, resources are renewable, and the objective is to minimize the project duration. A local search metaheuristic—tabu search is proposed to solve this strongly NP-hard problem, and it is compared with the multi-start iterative improvement method as well as with random sampling. A computational experiment is described, performed on a set of instances based on standard test problems constructed by the ProGen project generator. The algorithms are computationally compared, the results are analyzed and discussed, and some conclusions are given.     

A branch-and-price algorithm for scheduling parallel machines with sequence dependent setup times

We consider the problem of scheduling n independent jobs on m unrelated parallel machines with sequence-dependent setup times and availability dates for the machines and release dates for the jobs to minimize a regular additive cost function. In this work, we develop a new branch-and-price optimization algorithm for the solution of this general class of parallel machines scheduling problems. A new column generation accelerating method, termed “primal box”, and a specific branching variable selection rule that significantly reduces the number of explored nodes are proposed. The computational results show that the approach solves problems of large size to optimality within reasonable computational time.     

Variable neighbourhood search for redundancy allocation problems

^** Email: ycliang{at}saturn.yzu.edu.tw^*** Email: s929512{at}mail.yzu.edu.tw^**** Email: s927522{at}mail.yzu.edu.tw A variable neighbourhood search (VNS) algorithm has been developedto solve the redundancy allocation problem (RAP). The VNS methodis perfectly suited to those combinatorial problems with potentialneighbourhood structures, as in the case of the RAP. The moststudied configuration of the RAP is a series system of s-independentk-out-of-n:G subsystems the so-called series–parallelsystem. The RAP is to select the optimal combination and redundancylevels of components to meet system-level constraints. Two typesof objectives are considered in this study—system reliabilitymaximization and system cost minimization. The VNS algorithmis tested on sets of benchmark problems and compared to thebest heuristics in the literature such as tabu search, multipleweighted objective heuristic, ant colony optimization and geneticalgorithm. Computational results show the advantages and benefitsof VNS for solving both types of RAP while considering bothsolution quality and computational efficiency.     

Carryover sequence-dependent group scheduling with the integration of internal and external setup times

This paper addresses a group scheduling problem in a two-machine flow shop with a bicriteria objective and carryover sequence-dependent setup times. This special type of group scheduling problem typically arises in the assembly of printed circuit boards (PCBs). The objective is to sequence all board types in a board group as well as board groups themselves in a way that the objective function is minimized. We introduce the carryover sequence-dependent setup on machines, and call it internal setup. As an opportunity for manufacturers to decrease the costs, the focus is to completely eliminate the role of the kitting staff. Thus, we introduce the external setup (kitting) time for the next board group and require it to be performed by the machine operator during the time he is idle. Consequently, the internal and external setup times are integrated in this research, and to the best of our knowledge it is for the first time a research on PCB group scheduling is performed by integrating both setups. In order to solve this problem, first a mathematical model is developed. Then a heuristic together with two other meta-heuristic algorithms (one based on tabu search and the other based on genetic algorithm) are proposed and their efficiency and effectiveness on several problems are tested. Also a statistical experimental design is performed in order to evaluate the impact of different factors on the performance of the algorithms.     

A production scheduling heuristic for an electronics manufacturer with sequence-dependent setup costs

In this paper, we develop a three-step heuristic to address a production scheduling problem at a high volume assemble-to-order electronics manufacturer. The heuristic provides a solution for scheduling multiple product families on parallel, identical production lines so as to minimize setup costs. The heuristic involves assignment, sequencing, and time scheduling steps, with an optimization approach developed for each step. For the most complex step, the sequencing step, we develop a greedy randomized adaptive search procedure (GRASP). We compare the setup costs resulting from the use of our scheduling heuristic against a heuristic previously developed and implemented at the electronics manufacturer that assumes approximately equal, sequence-independent, setup costs. By explicitly considering the sequence-dependent setup costs and applying GRASP, our empirical results show a reduction in setups costs for an entire factory of 14–21% with a range of single production line reductions from 0% to 49%.     

Intelligent variable neighbourhood search for the minimum labelling spanning tree problem

Given a connected, undirected graph whose edges are labelled (or coloured), the minimum labelling spanning tree (MLST) problem seeks a spanning tree whose edges have the smallest number of distinct labels (or colours). In recent work, the MLST problem has been shown to be NP-hard and some effective heuristics have been proposed and analyzed. In a currently ongoing project, we investigate an intelligent optimization algorithm to solve the problem. It is obtained by the basic Variable Neighbourhood Search heuristic with the integration of other complements from machine learning, statistics and experimental algorithmics, in order to produce high-quality performance and to completely automate the resulting optimization strategy. Computational experiments show that the proposed metaheuristic has high-quality performance for the MLST problem and it is able to obtain optimal or near-optimal solutions in short computational running time.     

A variable neighbourhood search algorithm for the open vehicle routing problem

In the open vehicle routing problem (OVRP), the objective is to minimise the number of vehicles and then minimise the total distance (or time) travelled. Each route starts at the depot and ends at a customer, visiting a number of customers, each once, en route, without returning to the depot. The demand of each customer must be completely fulfilled by a single vehicle. The total demand serviced by each vehicle must not exceed vehicle capacity. Additionally, in one variant of the problem, the travel time of each vehicle should not exceed an upper limit.     

A hybrid model of integer programming and variable neighbourhood search for highly-constrained nurse rostering problems

This paper presents a hybrid multi-objective model that combines integer programming (IP) and variable neighbourhood search (VNS) to deal with highly-constrained nurse rostering problems in modern hospital environments. An IP is first used to solve the subproblem which includes the full set of hard constraints and a subset of soft constrains. A basic VNS then follows as a postprocessing procedure to further improve the IP’s resulting solutions. The satisfaction of the excluded constraints from the preceding IP model is the major focus of the VNS. Very promising results are reported compared with a commercial genetic algorithm and a hybrid VNS approach on real instances arising in a Dutch hospital. The comparison results demonstrate that our hybrid approach combines the advantages of both the IP and the VNS to beat other approaches in solving this type of problems. We also believe that the proposed methodology can be applied to other resource allocation problems with a large number of constraints.     

A multi-objective tabu search for a single-machine scheduling problem with sequence-dependent setup times

An m-objective tabu search algorithm for sequencing of n jobs on a single machine with sequence-dependent setup times is proposed. The algorithm produces a solution set that is reflective of the objectives’ weights and close to the best observed values of the objectives. We also formulate a mixed integer linear program to obtain the optimal solution of a three-objective problem. Numerical examples are used to study the behavior of the proposed m-objective tabu search algorithm and compare its solutions with those of the mixed integer linear program.     

Variable neighbourhood search for colour image quantization

^** Email: nenad.mladenovic{at}brunel.ac.uk Colour image quantization is a data compression technique thatreduces the total set of colours in a digital image to a representativesubset. This problem is first expressed as a large M-medianone. The advantages of this model over the usual minimum sum-of-squaresmodel are discussed first and then, the heuristic based on variableneighbourhood search metaheuristic is applied to solve it. Computationalexperience proves that this approach compares favourably withtwo other recent state-of-the-art heuristics, based on geneticand particle swarm searches.     

A hybrid heuristic ordering and variable neighbourhood search for the nurse rostering problem

This paper is concerned with the development of intelligent decision support methodologies for nurse rostering problems in large modern hospital environments. We present an approach which hybridises heuristic ordering with variable neighbourhood search. We show that the search can be extended and the solution quality can be significantly improved by the careful combination and repeated use of heuristic ordering, variable neighbourhood search and back-tracking. The amount of computational time that is allowed plays a significant role and we analyse and discuss this. The algorithms are evaluated against a commercial Genetic Algorithm on commercial data. We demonstrate that this methodology can significantly outperform the commercial algorithm. This paper is one of the few in the scientific nurse rostering literature which deal with commercial data and which compare against a commercially implemented algorithm.     

Group-shop scheduling with sequence-dependent set-up and transportation times

This paper considers a group-shop scheduling problem (GSSP) with sequence-dependent set-up times (SDSTs) and transportation times. The GSSP provides a general formulation including the job-shop and the open-shop scheduling problems. The consideration of set-up and transportation times is among the most realistic assumptions made in the field of scheduling. In this paper, we study the GSSP with transportation and anticipatory SDSTs, where jobs are released at different times and there are several transporters to carry jobs. The objective is to find a job schedule that minimizes the makespan, that is, the time at which all jobs are completed and transported to the warehouse (or to the customer). The problem is formulated as a disjunctive programming problem and then prepared in a form of mixed integer linear programming (MILP). Due to the non-deterministic polynomial-time hardness (NP-hardness) of the GSSP, large instances cannot be optimally solved in a reasonable amount of time. Therefore, a genetic algorithm (GA) hybridized with an active schedule generator is proposed to tackle large-sized instances. Both Baldwinian and Lamarckian versions of the proposed hybrid algorithm are then implemented and evaluated through computational experiments.     

A variable neighbourhood search for hybrid flow-shop scheduling problem with rework and set-up times

This paper deals with hybrid flow-shop scheduling problem with rework. In this problem, jobs are inspected at the last stage, and poorly processed jobs were returned and processed again. Thus, a job may visit a stage more than once, and we have a hybrid flow-shop with re-entrant flow. This kind of a shop may occur in many industries, such as final inspection system in automotive manufacturing. The criterion is to minimize the makespan of the system. We developed a 0–1 mixed-integer program of the problem. Since the hybrid flow-shop scheduling problem is NP-hard, an algorithm for finding an optimal solution in polynomial time does not exist. So we generalized some heuristic methods based on several basic dispatching rules and proposed a variable neighbourhood search (VNS) for the problem with sequence-dependent set-up times and unrelated parallel machines. The computational experiments show that VNS provides better solutions than heuristic methods.     

Balancing and scheduling tasks in assembly lines with sequence-dependent setup times

The classical Simple Assembly Line Balancing Problem (SALBP) has been widely enriched over the past few years with many realistic approaches and much effort has been made to reduce the distance between the academic theory and the industrial reality. Despite this effort, the scheduling of the execution of tasks assigned to every workstation following the balancing of the assembly line has been scarcely reported in the scientific literature. This is supposed to be an operational concern that the worker should solve himself, but in several real environments, setups between tasks exist and optimal or near-optimal tasks schedules should be provided inside each workstation. The problem presented in this paper adds sequence-dependent setup time considerations to the classical SALBP in the following way: whenever a task is assigned next to another at the same workstation, a setup time must be added to compute the global workstation time. After formulating a mathematical model for this innovative problem and showing the high combinatorial nature of the problem, eight different heuristic rules and a GRASP algorithm are designed and tested for solving the problem in reasonable computational time.     

Single-machine scheduling problems with past-sequence-dependent setup times

This paper studies single-machine scheduling problems with setup times which are proportionate to the length of the already scheduled jobs, that is, with past-sequence-dependent or p-s-d setup times. The following objective functions are considered: the maximum completion time (makespan), the total completion time, the total absolute differences in completion times and a bicriteria combination of the last two objective functions. It is shown that the standard single-machine scheduling problem with p-s-d setup times and any of the above objective functions can be solved in O(nlog n) time (where n is the number of jobs) by a sorting procedure. It is also shown that all of our results extend to a “learning” environment in which the p-s-d setup times are no longer linear functions of the already elapsed processing time due to learning effects.     

A bicriteria parallel machine scheduling with a learning effect of setup and removal times

In studies on scheduling problems, generally setup times and removal times of jobs have been neglected or by including those into processing times. However, in some production systems, setup times and removal times are very important such that they should be considered independent from processing times. Since, in general jobs are done according to automatic machine processes in production systems processing times do not differ according to process sequence. But, since human factor becomes influential when setup times and removal times are taken into consideration, setup times will be decreasing by repeating setup processes frequently. This fact is defined with learning effect in scheduling literature. In this study, a bicriteria m-identical parallel machines scheduling problem with a learning effect of setup times and removal times is considered. The objective function of the problem is minimization of the weighted sum of total completion time and total tardiness. A mathematical programming model is developed for the problem which belongs to NP-hard class. Results of computational tests show that the proposed model is effective in solving problems with up to 15 jobs and five machines. We also proposed three heuristic approaches for solving large jobs problems. According to the best of our knowledge, no work exists on the minimization of the weighted sum of total completion time and total tardiness with a learning effect of setup times and removal times.     

Grasp with memory-based mechanisms for minimizing total tardiness in single machine scheduling with setup times

This paper addresses the problem of scheduling jobs in a single machine with sequence dependent setup times in order to minimize the total tardiness with respect to job due dates. We propose variants of the GRASP metaheuristic that incorporate memory-based mechanisms for solving this problem. There are two mechanisms proposed in the literature that utilize a long-term memory composed of an elite set of high quality and sufficiently distant solutions. The first mechanism consists of extracting attributes from the elite solutions in order to influence the construction of an initial solution. The second one makes use of path relinking to connect a GRASP local minimum with a solution of the elite set, and also to connect solutions from the elite set. Reactive GRASP, which probabilistically determines the degree of randomness in the GRASP construction throughout the iterations, is also investigated. Computational tests for instances involving up to 150 jobs are reported, and the proposed method is compared with heuristic and exact methods from the literature.