In this article, we present a mixed-integer programming formulation for the simultaneous batching and scheduling in multiproduct multistage processes. The proposed sequence-based formulation addresses limitations of existing approaches where batching and scheduling decisions are carried out sequentially.
To account for batching decisions, we use additional batch-selection and batch-size variables and introduce demand-satisfaction and unit-capacity constraints. Assignment constraints are active only for the subset of batches that are selected, and sequencing is carried out between batches that are assigned on the same processing unit. We also propose an alternate formulation to handle sequence-dependent changeover costs. Finally, to enhance the computational performance of the model, we present methods that allow us to fix a subset of sequencing variables and we develop a class of tightening inequalities based on time windows.
View Author Information. Cite this: Ind. Article Views Altmetric -. Citations Abstract In this article, we present a mixed-integer programming formulation for the simultaneous batching and scheduling in multiproduct multistage processes.
How to use batching to improve SQL Database application performance
Supporting Information Available. Introduction 2. Background and Problem Description 3. Mathematical Formulation 4. A convenient pattern is to have a dummy Task at the end of several dependency chains, so you can trigger a multitude of pipelines by specifying just one task in command line, similarly to how e. This simple task will not do anything itself, but will invoke a bunch of other tasks.
Per each invocation, Luigi will perform as many of the pending jobs as possible those which have all their dependencies present. This class is used for tasks that only wrap other tasks and that by definition are done if all their requirements exist. A common requirement is to have a daily report or something else produced every night. Sometimes for various reasons tasks will keep crashing or lacking their required dependencies for more than a day though, which would lead to a missing deliverable for some date.
To ensure that the above AllReports task is eventually completed for every day value of date parameter , one could e. Then, so long as Luigi keeps being invoked, the backlog of jobs would catch up nicely after fixing intermittent problems. Luigi actually comes with a reusable tool for achieving this, called RangeDailyBase resp. Simply putting. NB - it will not always loop over everything from till current time though, but rather a maximum of 3 months ago by default - see RangeDailyBase documentation for this and more knobs for tweaking behavior.
See also Monitoring below. RangeDailyBase, described above, is named like that because a more efficient subclass exists, RangeDaily resp. RangeHourly , tailored for hundreds of task classes scheduled concurrently with contiguousness requirements spanning years which would incur redundant completeness checks and scheduler overload using the naive looping approach.
It has the same knobs as RangeDailyBase, with some added requirements. Also a common use case, sometimes you have tweaked existing recurring task code and you want to schedule recomputation of it over an interval of dates for that or another reason. Most conveniently it is achieved with the above described range tools, just with both start inclusive and stop exclusive parameters specified:. Some tasks you want to recur may include additional parameters which need to be configured. The Range classes provide a parameter which accepts a DictParameter and passes any parameters onwards for this purpose.
Alternatively, you can specify parameters at the task family level as described here , however these will not appear in the task name for the upstream Range task which can have implications in how the scheduler and visualizer handle task instances. One common way to do this is by simply running the maximum value. The minimisation of makespan arranges tasks in order to level the differences between the completion time of each work phase. It tries to smooth picks in work-centre occupancy to obtain batching in load assignment per time.
The possible representation of a JS problem could be done through a Gantt chart or through a Network representation. Gantt created innovative charts for visualizing planned and actual production [ 18 ]. According to Cox et al. Gantt designed his charts so that foremen or other supervisors could quickly know whether production was on schedule, ahead of schedule or behind schedule. A Gantt chart, or bar chart as it is usually named, measures activities by the amount of time needed to complete them and use the space on the chart to represent the amount of the activity that should have been done in that time [ 7 ].
A Network representation was first introduced by Roy and Sussman [ 20 ]. Tasks are defined in a network representation through a probabilistic model, observing the precedence constraints, characterized in a machine occupation matrix M and considering the processing time of each tasks, defined in a time occupation matrix T. The descriptions and notations as follow are due to Adams et.
V is a set of nodes representing tasks of jobs. C is the set of conjunctive arcs or direct arcs that connect two consecutive tasks belonging to the same job chain. These represent technological sequences of machines for each job;. In this representation all nodes are weighted with exception of source and sink node. A graph representation of a simple instance of JSP, consisting of 9 operations partitioned into 3 jobs and 3 machines, is presented in fig.
Let s v be the starting time of an operation to a node v.
- Luigi Patterns — Luigi documentation;
- Borel extension approach to weakly compact operators on C 0(T).
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By using the disjunctive graph notation, the JSPP can be formulated as a mathematical programming model as follows:. Disjunctive graph representation. There are disjunctive arcs between every pair of tasks that has to be processed on the same machine dashed lines and conjunctive arcs between every pair of tasks that are in the same job dotted lines. Job notation is used.
The second condition ensures time to start continuities.
In order to obtain a scheduling solution and to evaluate makespan, we have to collect all feasible permutations of tasks to transform the undirected arcs in directed ones in such a way that there are no cycles. While the total number of arcs, in job notation, is fixed considering the number of tasks and jobs of instance:. The number of arcs defines the possible combination paths.
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Each path from source to sink is a candidate solution for JSSP. The routing graph is reported in figure 2 :. A logic has to be implemented in order to translate the scheduling problem into an algorithm structure. Academic researches on scheduling problems have produced countless papers [ 23 ]. Scheduling has been faced from many perspectives, using formulations and tools of various disciplines such as control theory, physical science and artificial intelligence systems [ 24 ]. Criteria for optimization could be ranked from applying simple priority rules to determine which job has to be processed next at the work-centres i.
Guidelines in using heuristics in combinatorial optimization can be found in Hertz [ 27 ]. A classification of heuristic methods was proposed by Zanakis et al.
The first ones are focused on producing a solution based on an initial proposal, the goal is to decrease the solution until all the jobs are assigned to a machine, not considering the size of the problem [ 29 ]. The second ones are iterative algorithms which explore solutions by moving step by step form one solution to another.
The method starts with an arbitrary solution and transits from one solution to another according to a series of basic modifications defined on case by case basis [ 30 ].clublavoute.ca/pamim-donde-conocer.php
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Relatively simple rules in guiding heuristic, with exploitation and exploration, are capable to produce better quality solutions than other algorithms from the literature for some classes of instances. These variants originate the class of meta-heuristic approaches [ 31 ]. The meta-heuristics  - , and in general the heuristics, do not ensure optimal results but they usually tend to work well [ 32 ].
The purpose of the paper is to illustrate the most promising optimization methods for the JSSP. As optimization techniques, metaheuristics are stochastic algorithms aiming to solve a broad range of hard optimization problems, for which one does not know more effective traditional methods. Often inspired by analogies with reality, such as physics science, Simulated Annealing [ 33 ] and Electromagnetic like Methods [ 34 ], biology Genetic Algorithms [ 35 ], Tabu Search [ 36 ] and ethnology Ant Colony [ 37 ,], Bees Algorithm [ 38 ] , human science Neural Networks [ 39 ] , they are generally of discrete origin but can be adapted to the other types of problems.
The methodology of a GAs - based on the evolutionary strategy- trasforms a population set of individual objects, each with an associated fitness value, into a new generation of the population occurring genetic operations such as crossover sexual recombination and mutation fig. The theory of evolutionary computing was formalized by Holland in [ 40 ].
The theory of evolution is biologically explained, the individuals with a stronger fitness are considered better able to survive.. Cells, with one or more strings of DNA i.
The gene i. Physical manifestations are raised into genotype i. Each genotype has is physical manifestation into phenotype. According to these parameters is possible to define a fitness value. Combining individuals through a crossover i.
Related Batching and Scheduling: Models and Methods for Several Problem Classes
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