6 research outputs found
How the structure of precedence constraints may change the complexity class of scheduling problems
This survey aims at demonstrating that the structure of precedence
constraints plays a tremendous role on the complexity of scheduling problems.
Indeed many problems can be NP-hard when considering general precedence
constraints, while they become polynomially solvable for particular precedence
constraints. We also show that there still are many very exciting challenges in
this research area
Structural Properties of an Open Problem in Preemptive Scheduling
Structural properties of optimal preemptive schedules have been studied in a
number of recent papers with a primary focus on two structural parameters: the
minimum number of preemptions necessary, and a tight lower bound on `shifts',
i.e., the sizes of intervals bounded by the times created by preemptions, job
starts, or completions. So far only rough bounds for these parameters have been
derived for specific problems. This paper sharpens the bounds on these
structural parameters for a well-known open problem in the theory of preemptive
scheduling: Instances consist of in-trees of unit-execution-time jobs with
release dates, and the objective is to minimize the total completion time on
two processors. This is among the current, tantalizing `threshold' problems of
scheduling theory: Our literature survey reveals that any significant
generalization leads to an NP-hard problem, but that any significant
simplification leads to tractable problem.
For the above problem, we show that the number of preemptions necessary for
optimality need not exceed ; that the number must be of order
for some instances; and that the minimum shift need not be
less than . These bounds are obtained by combinatorial analysis of
optimal schedules rather than by the analysis of polytope corners for
linear-program formulations, an approach to be found in earlier papers. The
bounds immediately follow from a fundamental structural property called
`normality', by which minimal shifts of a job are exponentially decreasing
functions. In particular, the first interval between a preempted job's start
and its preemption is a multiple of 1/2, the second such interval is a multiple
of 1/4, and in general, the -th preemption occurs at a multiple of .
We expect the new structural properties to play a prominent role in finally
settling a vexing, still-open question of complexity
Some topics on deterministic scheduling problems
Sequencing and scheduling problems are motivated by allocation of limited resources over time. The goal is to find an optimal allocation where optimality is defined by some problem specific objectives.
This dissertation considers the scheduling of a set of ri tasks, with precedence constraints, on m \u3e= 1 identical and parallel processors so as to minimize the makespan. Specifically, it considers the situation where tasks, along with their precedence constraints, are released at different times, and the scheduler has to make scheduling decisions without knowledge of future releases. Both preemptive and nonpreemptive schedules are considered. This dissertation shows that optimal online algorithms exist for some cases, while for others it is impossible to have one. The results give a sharp boundary delineating the possible and the impossible cases.
Then an O(n log n)-time implementation is given for the algorithm which solves P|pj = 1, rj, outtree| ΣCj and P|pmtn, pj=1,rj,outtree|ΣCj.
A fundamental problem in scheduling theory is that of scheduling a set of n unit-execution-time (UET) tasks, with precedence constraints, on m \u3e 1 parallel and identical processors so as to minimize the mean flow time. For arbitrary precedence constraints, this dissertation gives a 2-approximation algorithm. For intrees, a 1.5-approximation algorithm is given.
Six dual criteria problems are also considered in this dissertation. Two open problems are first solved. Both problems are single machine scheduling problems with the number of tardy jobs as the primary criterion and with the total completion time and the total tardiness as the secondary criterion, respectively. Both problems are shown to be NP-hard. Then it focuses on bi-criteria scheduling problems involving the number of tardy jobs, the maximum weighted tardiness and the maximum tardiness. NP-hardness proofs are given for the scheduling problems when the number of tardy jobs is the primary criterion and the maximum weighted tardiness is the secondary criterion, or vice versa. It then considers complexity relationships between the various problems, gives polynomial-time algorithms for some special cases, and proposes fast heuristics for the general case
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The scheduling of manufacturing systems using Artificial Intelligence (AI) techniques in order to find optimal/near-optimal solutions.
This thesis aims to review and analyze the scheduling problem in general and Job Shop Scheduling Problem (JSSP) in particular and the solution techniques applied to these problems. The JSSP is the most general and popular hard combinational optimization problem in manufacturing systems. For the past sixty years, an enormous amount of research has been carried out to solve these problems. The literature review showed the inherent shortcomings of solutions to scheduling problems. This has directed researchers to develop hybrid approaches, as no single technique for scheduling has yet been successful in providing optimal solutions to these difficult problems, with much potential for improvements in the existing techniques.
The hybrid approach complements and compensates for the limitations of each individual solution technique for better performance and improves results in solving both static and dynamic production scheduling environments. Over the past years, hybrid approaches have generally outperformed simple Genetic Algorithms (GAs). Therefore, two novel priority heuristic rules are developed: Index Based Heuristic and Hybrid Heuristic. These rules are applied to benchmark JSSP and compared with popular traditional rules. The results show that these new heuristic rules have outperformed the traditional heuristic rules over a wide range of benchmark JSSPs. Furthermore, a hybrid GA is developed as an alternate scheduling approach. The hybrid GA uses the novel heuristic rules in its key steps. The hybrid GA is applied to benchmark JSSPs. The hybrid GA is also tested on benchmark flow shop scheduling problems and industrial case studies. The hybrid GA successfully found solutions to JSSPs and is not problem dependent. The hybrid GA performance across the case studies has proved that the developed scheduling model can be applied to any real-world scheduling problem for achieving optimal or near-optimal solutions. This shows the effectiveness of the hybrid GA in real-world scheduling problems.
In conclusion, all the research objectives are achieved. Finaly, the future work for the developed heuristic rules and the hybrid GA are discussed and recommendations are made on the basis of the results.Board of Trustees, Endowment Fund Project, KPK University of Engineering and Technology (UET), Peshawar and Higher Education Commission (HEC), Pakista