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Some concluding remarks and directions for future work are provided in Section 6. The multisystem optimization problem studied in this paper is derived from a textile printing and dyeing plant.
Its main processes include dyeing, printing, and finishing, and each process consists of multiple procedures. For example, the dyeing process includes singeing, bleaching, mercerizing, setting, and so on. According to raw materials and fabric structures, the process flow is not a completely fixed model for specific printing and dyeing requirements. Meanwhile, with the rapid development of manufacturing, resource saving and environmental protection have been increasingly concerned.
Textile printing and dyeing is an energy- and resource-intensive manufacturing process and uses physical and chemical methods to make products out of the raw textile, with extensive electricity and water as the treatment mediums. Therefore, it is urgent for manufacturers to improve production efficiency while reducing electricity consumption and water wastage. In this section, we propose an integrated optimization model for textile printing and dyeing by considering both production efficiency and resource saving.
Figure 1 shows a workflow of textile printing and dyeing processes, where the model includes a production scheduling subsystem based on production orders and a resource saving subsystem based on environment protection standard. These two subsystems are coupled with the other and share partial variables and constraints.
The whole model is a hybrid flow-shop scheduling problem with unrelated parallel machines, and it is a nondeterministic polynomial hard NP-hard problem. In the following, we will focus on a mathematical formulation of this integrated optimization model. For convenience, the symbols and notations used in the model are shown in Table 1. For a production scheduling subsystem in textile printing and dyeing, its mathematic model is formulated as a multiobjective optimization problem.
Suppose the subsystem schedules a set of jobs , and each job is completed orderly by procedures. Each procedure has unrelated parallel machines, and each job is worked on one machine in a procedure. A set-up time is required before a procedure of a job is processed on a machine.
The production scheduling subsystem has two conflicting objectives of makespan and production cost, which should be minimized, respectively. This subsystem is defined as follows:. In equation 1 , denotes the production scheduling subsystem, including makespan and production cost. In equation 2 , is the finishing time of the job through all procedures, which is iteratively calculated by where is the processing time of the procedure of the job running on the machine and is the finishing time of the job through the procedure.
In equation 3 , is production cost of the procedure of the job running on the machine. If the procedure of the job is processed by the machine, then ; otherwise,. For a resource saving subsystem in textile printing and dyeing, electricity consumption, water wastage, and processing cost caused by these components are the most important factors, which vary with the process sequences.
The mathematic model of this subsystem is also formulated as a multiobjective optimization problem, including three conflicting objectives of electricity consumption, water wastage, and processing cost, which should be minimized, respectively.
The subsystem is defined as follows:. In equation 5 , denotes the resource saving subsystem, including electricity consumption , water wastage , and the corresponding processing cost. Electricity consumption denoted by equation 6 is composed of four parts: processing electricity consumption, set-up electricity consumption, standby electricity consumption, and auxiliary electricity consumption.
The first term in equation 6 represents processing electricity consumption, where is the unit processing power of the machine when processing the procedure of the job.
The second term represents set-up electricity consumption, where and are, respectively, the set-up time and unit power of the machine when the procedure changes to the procedure for the job. The third term represents standby electricity consumption, where and are, respectively, the standby time and unit power of the machine when the procedure changes to the procedure for the job. The fourth term represents auxiliary electricity consumption of the auxiliary equipments in the machining process, which is relevant to the production time and auxiliary power, with the former consisting of processing time, set-up time, and standby time, and is the auxiliary unit power of the machine in the procedure.
Equation 7 denotes water wastage, which is different from electricity consumption. It considers two types of water wastage, including processing water wastage and set-up water wastage, and water wastage for auxiliary equipments and the standby stage of machines is neglected. In equation 7 , the first term represents processing water wastage, and is the unit water wastage of the machine when processing the procedure of the job. The second term represents set-up water wastage, and is the unit water wastage of the machine in the set-up stage of the procedure.
Equation 8 denotes the total processing cost, including raw material cost, electricity consumption cost, water consumption cost, and so on. Furthermore, for the proposed integrated optimization model of textile printing and dyeing, the solutions must satisfy the following constraints. Each machine handles exactly one procedure in a job: and each procedure of each job is processed in only one machine sometime:. In the above constraints, equation 9 is stand for assignment of one procedure in only one job to a machine, and equation 10 is stand for assignment of each procedure of each job to only one machine.
Now, an integrated optimization model for textile printing and dyeing has been presented, and the following section develops a multisystem optimization algorithm that will be used to solve the proposed optimization problem. This section first presents a MSO framework, which can serve as a template for extending any other heuristic methods to the MSO algorithm.
Then, it presents the implementation of the proposed MSO algorithm. A MSO problem in this paper consists of multiple subsystems, which are coupled with the others and share partial objectives or constraints. That is, each subsystem in a MSO problem has not only coupled objectives and constraints but also independent objectives and constraints. Therefore, multisystem optimization is more complicate than traditional multiobjective optimization.
Suppose that we have a complex system that consists of several subsystems. Without loss of generality, all subsystems are assumed to be minimization problems. Inspired by implicit parallelism of multipopulation heuristic approaches [ 19 , 20 ], we first treat a subpopulation as a subsystem to optimize a subsystem optimization problem by excellent evolution operators.
Then, we realize information sharing between multiple subsystems by migration, based on the relations of sharing variables and similarity levels between objectives and constraints, to accelerate the global optimization of the whole system.
Based on the above idea, each subsystem is comprised of three sets of elements. The first set includes candidate solutions to the subsystem optimization problem. The second and third sets include the objectives and constraints of the subsystem.
The MSO algorithm mainly includes two steps: evolution within subsystems and sharing information via migration across subsystems.
We refer to these two types of operators as within-subsystem evolution and cross-subsystem migration. The MSO framework is illustrated in Figure 2 , where a complex multisystem problem includes multiple coupled subsystems, and each subsystem includes multiple objectives and constraints.
Within-subsystem evolution is used in each subsystem, and cross-subsystem migration is used between multiple subsystems. In the proposed MSO architecture, we use a modified version of NSGA-II [ 21 ], initially designed for single systems with multiobjectives, as a within-subsystem evolution operator. The modified NSGA-II employs solution ranks as selection probabilities considering the relative performance of a candidate solution because each subsystem has its own set of candidate solutions, objectives, and constraints, and the ranks assigned to the candidate solutions in a subsystem denote the relative fitness of those solutions only in that particular subsystem.
Then, we recombine the candidate solutions using any desired recombination method in heuristic methods. Finally, we mutate the child population and replace the parents with the children. Cross-subsystem migration is an important operator in the MSO algorithm. For the development of heuristic methods including the proposed MSO algorithm in this paper, we must consider two challenges.
One is to converge to the optimal solutions. To address this challenge in the MSO algorithm, we define similarity levels for both objectives and constraints. If two subsystems have high similarity levels, the optimization problems of those subsystems are similar to each other. This also means that the features that are important in one subsystem have a similar level of importance in the other subsystem. Migration between subsystems with similar objectives and constraints is expected to be helpful for all such subsystems.
Another important challenge is to maintain population diversity as the main factor that enables the population to improve. If the population has a low diversity, most of candidate solutions are similar to each other, and the probability that a candidate solution improves after migration is low.
In this case, migration may not effectively contribute to improvement in the population. Medicare for Health Maintenance The training of the best athletes places foundational health first. This is especially important for Senior Games participants 65 and older, the largest segment of our athlete population. Receiving basic preventive care for health maintenance is key to foundational health. Open Enrollment is your chance to review and compare your current Medicare coverage.
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