Semi-active Schedules Research Articles

Stability of a schedule minimising the makespan for processing jobs on identical machines

ABSTRACT A set of jobs has to be processed on identical machines. Every job may be processed on any available machine without preemptions. The criterion is to minimise the makespan (i.e. the completion time of the last job in a schedule). During the realisation of a schedule, durations of some jobs may deviate from the initial values estimated before scheduling. Other jobs have fixed durations that are known before scheduling. We conduct a stability analysis of the optimal semi-active schedule. First, we derive necessary and sufficient conditions for an optimal schedule to be unstable with respect to infinitely small variations of the non-fixed durations (the stability radius of an unstable schedule is equal to zero). Second, we show that the stability radius of an optimal schedule could be infinitely large. Furthermore, several lower and upper bounds on the stability radius have been established. Third, we derive a formula and develop an algorithm for calculating stability radii.

Inventory and total completion time minimization for assembly job-shop scheduling considering material integrity and assembly sequential constraint

Assembly job-shop scheduling problems (AJSSP), considering material integrity and assembly sequential constraint (AJSSP-IS), exist widely in rolling stock industries. Production efficiency and inventory levels are the key concerns for this type of industry. Thus, this paper investigates an AJSSP-IS via a lexicographical mixed-integer linear programming (MILP) model and an enhanced simulated annealing algorithm (ESA) to minimize a regular primary objective, the total completion time and an unregular secondary objective, the total inventory time. The lexicographical MILP model is formulated to determine the processing sequence and assembly sequence, and a lexicographical optimization method is applied to obtain the best scheme of a secondary objective from the optimal schemes of a primary objective. Further, an ESA is developed to solve the problem efficiently. In ESA, an assembly-driven initialization is designed to create high-quality solutions by considering assembly relationships; a two-stage decoding is developed in which a semi-active schedule achieves the minimization of the total completion time and a right-shift schedule reduces the total inventory time without extending the total completion time; an improved neighborhood search is proposed by introducing a critical block-oriented selection and a filter selection to increase search efficiency. Experimental results indicate that the lexicographical MILP model can find the lower bound of small-scale problems, and the ESA significantly outperforms six state-of-the-art algorithms in fixing the considered problem.

Efficiency of the solution representations for the hybrid flow shop scheduling problem with makespan objective

In this paper we address the classical hybrid flow shop scheduling problem with makespan objective. As this problem is known to be NP-hard and a very common layout in real-life manufacturing scenarios, many studies have been proposed in the literature to solve it. These contributions use different solution representations of the feasible schedules, each one with its own advantages and disadvantages. Some of them do not guarantee that all feasible semiactive schedules are represented in the space of solutions –thus limiting in principle their effectiveness– but, on the other hand, these simpler solution representations possess clear advantages in terms of having consistent neighbourhoods with well-defined neighbourhood moves. Therefore, there is a trade-off between the solution space reduction and the ability to conduct an efficient search in this reduced solution space. This trade-off is determined by two aspects, i.e. the extent of the solution space reduction, and the quality of the schedules left aside by this solution space reduction. In this paper, we analyse the efficiency of the different solution representations employed in the literature for the problem. More specifically, we first establish the size of the space of semiactive schedules achieved by the different solution representations and, secondly, we address the issue of the quality of the schedules that can be achieved by these representations using the optimal solutions given by several MILP models and complete enumeration. The results obtained may contribute to design more efficient algorithms for the hybrid flow shop scheduling problem.

Mixed graph model and algorithms for parallel-machine job-shop scheduling problems

Heuristic algorithms are developed to solve the parallel-machine job-shop problems , where the criterion is the minimisation of the makespan, , or the sum of completion times, . The developed algorithms include sequencing and assigning stages. At the sequencing stage, the job-shop problem is solved, where job is available for processing from time . The problem is modelled by a mixed graph , where the vertices are the operations to be processed. The precedence constraints on the set are determined by the arc set . The resource constraints are determined by the edge set . In order to resolve a conflict arising between two operations processed on the same machine, the algorithm should substitute a conflict edge from the set by an arc incident to the same vertices from the set . The resulting digraph determines a heuristic solution to the problem , where all machines are different. The digraph determines a semi-active schedule for the problem . A mixed graph model is used for solving the problem , which allows a scheduler to construct an efficient schedule via deleting some arcs from the set in the digraph or (and) via changing orientations of the arcs. Several heuristics have been developed to transform the digraph into a new digraph as a proper answer for the problem . The developed algorithms have been tested on the benchmark instances. It is demonstrated how these algorithms may be used for solving a train timetabling problem.

A synergetic combination of small and large neighborhood schemes in developing an effective procedure for solving the job shop scheduling problem.

This paper presents an effective procedure for solving the job shop problem. Synergistically combining small and large neighborhood schemes, the procedure consists of four components, namely (i) a construction method for generating semi-active schedules by a forward-backward mechanism, (ii) a local search for manipulating a small neighborhood structure guided by a tabu list, (iii) a feedback-based mechanism for perturbing the solutions generated, and (iv) a very large-neighborhood local search guided by a forward-backward shifting bottleneck method. The combination of shifting bottleneck mechanism and tabu list is used as a means of the manipulation of neighborhood structures, and the perturbation mechanism employed diversifies the search. A feedback mechanism, called repeat-check, detects consequent repeats and ignites a perturbation when the total number of consecutive repeats for two identical makespan values reaches a given threshold. The results of extensive computational experiments on the benchmark instances indicate that the combination of these four components is synergetic, in the sense that they collectively make the procedure fast and robust.

Scheduling multiprocessor tasks to minimise the makespan on two dedicated processors

In this paper, we consider the problem of scheduling tasks on two dedicated processors where some tasks need to be processed only by the first processor and some others by the second processor; the remaining tasks, however, need both processors simultaneously. Tasks have release dates and have to be scheduled without preemption. The objective is to minimise its makespan. For this problem, which is known to be NP-hard in the strong sense, we propose a lower bound based on processor relaxation and show that it is equal to the optimal solution for a preemptive case. We propose two heuristics with a worst-case analysis and its generalisation to any semi-active schedule. A set of dominance properties are established and a branch-and-bound algorithm is developed and tested on a set of randomly generated instances. [Received 29 September 2008; Revised 01 March 2009; Accepted 13 April 2009]

Flexible solutions in disjunctive scheduling: General formulation and study of the flow-shop case

We consider the context of decision support for schedule modification after the computation off-line of a predictive optimal (or near optimal) schedule. The purpose of this work is to provide the decision maker a characterization of possible modifications of the predictive schedule while preserving optimality. In the context of machine scheduling, the anticipated modifications are changes in the predictive order of operations on the machines. To achieve this goal, a flexible solution feasible w.r.t to operations deadlines, is provided instead of a single predictive schedule. A flexible solution represents a set of semi-active schedules and is characterized by a partial order on each machine, so that the total order can be set on-line, as required by the decision maker. A flexible solution is feasible if all the complete schedules that can be obtained by extension are also feasible. In this paper we develop two main issues. The first one concerns the evaluation of a flexible solution in the worst case allowing to certify if the solution is feasible. The second issue is the computation of feasible (w.r.t deadlines) flexible solutions of maximal flexibility imposed by the decision maker. Under an epsilon -constraint framework, solving this problem allows to find compromise solutions for the flexibility criterion and any minmax regular scheduling criterion. The special case of the flow-shop scheduling problem is studied and computational experiments are carried out.

Active and stable project scheduling

The paper proposes a new classification of schedules for resource-constrained project scheduling problems with minimum and maximum time lags between project activities and regular and different types of nonregular objective functions. The feasible region of the scheduling problems represents a (generally disconnected) union of polytopes. In addition to the well-known concepts of active and semiactive schedules, pseudoactive and quasiactive as well as stable, semistable, pseudostable, and quasistable schedules are introduced. The (quasi-, pseudo-, semi-)active schedules are related to different types of left-shifts of sets of activities and correspond to minimal points of certain subsets of the feasible region. The (quasi-, pseudo-, semi-)stable schedules do not allow oppositely directed shifts and correspond to extreme points of certain subsets of the feasible region. The different sets of schedules contain optimal schedules for project scheduling problems which differ in their objective functions. The correspondence between those sets of schedules and vertices of specific polyhedral subsets of the feasible region can be exploited for analyzing schedule generation schemes which have been developed recently for finding solutions to the different classes of project scheduling problems.

Note: On Semi-Active Timetabling in Resource-Constrained Project Scheduling

The purpose of this note is to demonstrate that the branch-and-bound algorithm presented in Demeulemeester and Herroelen (1992) does not reduce the enumeration to semi-active schedules.

Scheduling with Sequence-Dependent Setup Times and Early-Tardy Penalties

This paper deals with the single machine scheduling problem under the assumption that the machine requires a setup between jobs and that the setup times are sequencedependent. The objective function is to minimize the weighted sum of setup, tardiness and earliness costs. A tabu search-based heuristic is proposed which incorporates a methodology for the generation of semi-active schedules, which includes idle time between jobs. The approach allows simultaneous determination of the sequencing and scheduling of the jobs.

S3 sets. An extension of the Beale-Tomlin special ordered sets

In this work an extension of the Beale-Tomlin special ordered sets is introduced that has proved to be efficient for solving certain types of open shop scheduling problems. Besides their usual characteristics, exclusivity constraints in the jobs are allowed, more general than tree-like precedence structures are considered, and semi-active schedules that cannot be labeled as non-optimal solutions may occur. The problem is formulated as a large-scale 0–1 model. Computational experience on some real-life problems is reported.