Research on Scheduling Problem of Manufacturing/remanufacturing Hybrid Systems

Remanufacturing plays a significant effect on saving social resources, developing green economy and reducing enterprise cost. Aiming at a production scheduling problem in manufacturing/remanufacturing hybrid systems we investigated, a multi-objective optimal scheduling model is built. The goals of optimization are to minimize total equipment idle time, total delivery delay and total setup time which are consistent with actual needs of the enterprise. An improved NSGA-II is adopted to increase the population diversity and improve the search performance. The similarity degree S is employed to evaluate the diversity of population in this paper. Crossover and mutation operations are adjusted adaptively based on S. This algorithm is applied to an engine manufacturing enterprise compared with the original genetic algorithm. The analysis of experimental results shows that the way in this paper has certain superiority.


Introduction
The activities of remanufacturing can economize the costs of production and efficiently lower the pollution to the environment, which is a significant way to sustainable development and realize cyclic economy. The process of remanufacturing has production links, some uncertain factors and complicated cooperation relationships compared with the traditional manufacturing process, making it tough for the traditional production scheduling optimization method to satisfy the demands of the remanufacturing production process.
Many scholars and experts come up with different solutions to the remanufacturing scheduling problem and made contributions to this theme. Polotskia et al. [1]thought over the switch of equipment production mode. In order to satisfy the various needs of customers for products, they optimized the switching process of equipment production mode to minimize the total cost. Giglio et al. [2] comprehensively optimized batch size and workshop efficiency in a hybrid manufacturing/ remanufacturing system that produces multiple products. Lou et al. [3] researched the production capacity constraints in manufacturing/remanufacturing system, and established three batch decision models under different conditions. Liu et al. [4]had a research on the remanufacturing process scheduling problem in an inconclusive environment, and put forward the corresponding compounded intelligent algorithm to solve it.
In the manufacturing/remanufacturing hybrid production system, the multi-objective optimization is more in line with the practical needs. On the grounds of the features of the problem, this paper optimizes based on NSGA-Ⅱ, and uses similarity to improve the genetic operator.

Problem description
A company's engine component production line is a hybrid manufacturing/remanufacturing system that can produce both new products and re-products. One equipment corresponds to a process of product processing, and the processing of all products needs to flow in a single direction in which the equipment is arranged. According to process requirements, new products are processed in accordance with the conventional process route (process1→process2→process 3→…→process m), while the processing of remanufactured products needs to skip some procedures in the conventional process route and directly enter the later process for processing. However, the flow direction of components during processing remains consistent with the conventional process route.
The main features of the manufacturing/remanufacturing line are: (1) Each equipment needs a preparation time when it is converted from a new product manufacturing method to a remanufacturing method and vice versa; (2) Each process has an enough buffer zone, and only after one part is completed can another part be processed; (3) The equipment used in each process is fixed, and each process has only one equipment.
The scheduling needs to decide the processing sequence of the parts to be processed on each equipment (the start and end time of each component on each piece of equipment). The optimization goal is to put off the delivery of the total parts and minimize the total equipment vacant time, on the grounds of the requirements of the company. Minimum period and minimum total equipment setup time.

Mathematical model
The parameters used in this article is shown as follows.
(1) N: total number of products; (2) M: total number of equipment; (3) tij: processing time of the i th products on the j th equipment; (4) bj: number of parts processed on the j th equipment; (5) ts: equipment preparation time when the method is changed; (6) rk: the mode of production of the k th part on the j th equipment, k= 1,2..., bj. when produced as a new product, rkj= 0; otherwise, rkj= 1; (7) di: delivery date of part i; (8) tskj: the start time of the k th machined part on the j th equipment, k= 1,2..., bj; (9) tfkj: the completion time of the k th machined part on the j th equipment; (10) Oj: the j th equipment 's occupancy time; (11) Pj: the total time of the actual machining parts of the j th equipment.
Definition 1 Equipment occupancy time Oj refers to the total time span occupied by the equipment from the processing of the first part of the equipment to the last part (i.e. bj) assigned by the equipment.

Definition 2
The idle time of equipment refers to the time during which no parts are being processed during the occupied time of the equipment. It (the j th equipment) should be (Oj-Pj) .
The optimization model can be described as follows:

Coding
Individual coding adopts the serial number full arrangement and combination of the processing sequence of parts on the first equipment. For example, individual (1,4,3,2) means that parts are processed in the sequence of 1-4-3-2.

Crossover and mutation operations
This paper adopts the sequential crossover (OX) method [5]to generate offspring. Two crossover points are randomly generated from parent generation 1and the genes between them maintain the same position to inherit to offspring 1. Look for genes that are not in the crossover points of the parent 1 but in parent 2, and insert the position of the missing gene in the offspring 1 in sequence. The crossover mode is shown in Figure 1. Mutation operation is of great significance for jumping out of local optimum. In this paper, two mutation points were randomly generated and two genes were exchanged for mutation. The mutation mode is shown in Figure 2.

Adaptive crossover and mutation probabilities based on similarity degree
With the continuous evolution of the population, the similarity of scheduling schemes increases, which makes the algorithm easy to fall into local optimal. In this paper, similarity degree S is defined to measure the diversity of the population. When S is small, the population is relatively dispersed, decreasing and ; When S is large, the population tends to be uniform, increasing and . and can be adjusted adaptively to avoid the rapid convergence caused by the continuous crossover and reproduction of similar individuals [6].    The similarity of individuals in the population is:

Property 1 The maximum value of S is 1 for n≤ N.
Demonstration When all the individuals of the population are exactly the same, the maximum degree of similarity is achieved. At this point, End. Property 2 When n≤ N, if is an integer, the minimum value of S is 1 ; otherwise, it is slightly greater than 1 .

End.
Therefore, the calculation formula of crossover probability and mutation probability based on similarity degree is as follows: (3.8) (3.9)

Algorithm steps
The algorithm flow of adaptive NSGA-Ⅱ is shown in Figure 3.
(2) Uniform Index SP. Smaller SP means the distribution of solutions is more uniform.

Experimental results
The parameters of the algorithm are as follows: the population size is 60. For improved NSGA-Ⅱ, 0.9 is value of the maximum cross probability, 0.1 is value of the minimum cross probability, 0.1 is value of the maximum mutation probability, 0.01 is value of the minimum mutation probability. For NSGA-Ⅱ, the crossover probability is 0.9 and the mutation probability is 0.1. The algorithm iterates 300 times, and each group of experiments is run for 20 times. Each index is taken the average value. The algorithm program was programmed in Python. The simulator runs on a Windows 10 operating system computer with a 1.80GHz Intel (R) Core (TM) CPU. Table 2 presents the algorithm results.