# A Scheduling Solution for Robotic Arm-Based Batching Systems with Multiple Conveyor Belts +
## 5 Computational Burden of the Proposed Solution Approach  
Recall that a conveyor belt moves fast, feeding items to robotic arms at the same speed. Therefore, considering the fact that it is almost infeasible to know when and how food items enter into a robotic arm-based batching system, solving the robotic arm scheduling problem should be done in (near) real time.  
To understand the practical scale of a food processing system and the corresponding computational requirements for the proposed solution approach, we took a field trip to a food processing company in Denmark and conducted interviews with employees of the company. From the activity, we conclude that a schedule for an item should be derived before the item passes the buffer zone that spans from the entry point of the conveyor belt to the entry point of the robotic arms' operational area. The travel time for an item to pass the buffer zone is set as 3.6 s based on the interview. Because items keep entering into a conveyor belt, making it difficult for a scheduling algorithm to fully use the 3.6 s to solve the problem, we assume that only half of the buffer passing time, 1.8 s, is allowed for the scheduling algorithm to solve the scheduling problem. In other words, a series of sub-problems, each of which schedules multiple robotic arms for a specific tray, should be solved within 1.8 s.  
To examine the feasibility of the proposed solution approach in the above mentioned context, we develop a simulation model for a food batching system with two robotic arms. The layout of the target food batching system is illustrated in Figure 4(a), which is also used to visualize the simulation as shown in Figure 4(b). In the figure, the operational area for the robotic arms is colored in green. The buffer zone, consisting of 18 fields, starts from the left side of the conveyor belt, and an item just registered takes 3.6 s to pass the buffer zone. Considering a practical arrival rate of food items at the system, the mean inter-arrival time of items is set as 1.2 s, sampled from exponential distribution. The simulation runs for 30 min in a single replication, and a sub-problem is solved using CPLEX. The experiments are run on a Windows 10 machine with an Intel Core i5-8265U 4-core CPU @ 1.6 GHz with 8 GB RAM.  
Figure 6 shows a histogram of the computation time spent to solve a series of the sub-problems for a food item batch (i.e., the items in the first half of the buffer zone). In the chart, 1116 batches are considered. The figure shows that 92% of schedules are derived in less than 1.8 s (the dotted line). The median computational time is 0.66 s with a standard deviation of 0.69.  
Figure 5: A snapshot of the implemented simulation model.  
It should be pointed out that the success rate of 92% is satisfactory, but it tells us that one out of ten scheduling attempts will require more time than allowed, degrading the performance of the robotic arm-based system. A potential approach to tackle this issue is to consider local search heuristics, which are simple but effective for many combinatorial decision-making problems [29].