Considering the page limitation and the protection of the project, in this section, only a simplified case study will be used to show the modeling, the solving methods, and the results.
4.1 Case study
The layout of the case study is the same as this one shown in Figure 1.
Suppose there are driving direction limitations on some ways. After considering these limitations and the production flow, two driving ways are identified, as shown in Figure 2, which are marked in white. There are two routes one each way. Therefore, there are altogether four routes as predefined routes. Correspondingly there are four 0-1 binary variables. The destinations or PoUs with numbers under the unit load symbols and material requirement amount in small loading containers per hour are also shown in Figure 2. It can also be seen which destinations can be reached by which route. The required other parameters for the milk-run are given in Table 1. Based on the information, the transportation time for each route can be calculated. For route 1 and route 2, seven minutes are needed for the tugger train. For route 3 and route 4, five minutes are required.
Route 1: 840 m
m: Material demand in SLCs/h at this destination Source
Table 1: Input relating to tugger train for milk-run.
Speed of tugger train: 2 m/s Capacity of tugger train: 24 SLCs Time to load the trailers: 2 min Time to exchange SLCs: 0.5 min Time to unload the empties: 0.2 min Waiting time at each stop: 0.2 min
4.2 Optimization methods
For the optimization problem introduced in section 3, two methods have been developed. One method uses a specific heuristic “best-fit” in traduced in the paper from Klenk et al. [1]. One destination is assigned to the tugger train on one route, if the capacity of the tugger train is reached, then this destination will be assigned to the next route. If the capacity of the tugger train is not reached, then this destination is assigned to the tugger train on this route. Continuously, the next destination will be handled in the same way. With this heuristic the demand can be evenly distributed [1].
The second method being tested bases on genetic algorithm, exactly speaking, cooperative coevolutionary algorithm (CCEA). This algorithm has been successfully implemented in the research from Li et al. for the optimization [17]. In the method with CCEA for milk-run, there are two separated populations defined. One population is for the route and the other population is composed of the solutions for the assignment of destination to routes and the interval. These two populations evolve separately almost for the whole process. Only for the fitness, they both have to build the connection, because the fitness of one complete solution is influenced by all of the variables.
4.3 Results
The performance of the model for milk-run and the methods have been tested based on different examples and case studies regarding the performance. The best results for the simplified case study in the first subsection are shown in Table 2.
Table 2: Results of the example.
Route 1 Route 2 Route 3 Destinations to be supplied 1, 2, 5 3, 8 7, 4, 6
Interval (min) 30 30 30
Cycle time of one tour (min) 26.4 24.8 23
Transport time of one tour (min) 7 7 5
SLCs exchange time of one tour (min) 12 11 11 Empty unloading time of one tour (min) 4.8 4.4 4.4
Waiting time of one tour (min) 0.6 0.4 0.4
Trailer loading time of our tour (min) 2 2 2
The results show that the method with cooperative coevolutionary algorithm is always stable comparing to the other one. It is because the decisions of route selection and destination assignment to routes are not on the same level. The cooperative coevolutionary algorithm is just proper for this kind of characteristic. There are three routes selected and the optimal interval for them equals 30 minutes. The three routes need almost the same cycle time for tours. Based on the tact time of 30 minutes, no tugger trains can be used for two routes. Hence, altogether three tugger trains are required.
5 CONCLUSION
This paper is motivated by a project of implementing in-plant milk-run system in a company for the production and assembly line. Based on the extensive literature, the milk-run concept was at first proposed, in which the basic layout is fixed, the workstations are clearly defined, the activities or steps included in the milk-run process are described. Undoubtedly, for this concept the material requirements at each workstation or point of use are already known. As for the capacity, limited by the area, it is better to tow two trailers for a better steering and higher transport speed. Based on all these preconditions, constraints and partial solution techniques, optimization methods of using specific heuristic and genetic algorithm have been developed and compared for the best solution. Although the economic aspect has not so much improved, the milk-run concept brings mainly the benefit of higher standardization and place utilization.
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