## 4 Multi-Objective Optimization for Edible Oil Transportation Scheduling Method
### Pareto Front  
When numerous feasible solutions exist within the objective space, the Pareto front represents the boundary of solutions that satisfy all constraints, as illustrated by the red line in Figure 6. This boundary represents the set of all potentially optimal solutions in a multi-objective optimization problem. In this study, two key evaluation criteria, pipeline resource utilization and edible oil transportation time, are used to compare and analyze various task scenarios. For instance, as shown in Figure 6, although Task A consumes fewer pipeline resources (_f_2), it requires a longer transportation time (_f_1) compared to Task B. Conversely, Task B utilizes more resources but achieves a shorter transportation time. Achieving an optimal balance between transportation time and equipment usage is critical, as it directly influences the overall efficiency and performance of the scheduling system.  
In pipeline scheduling for edible oil transportation, multi-objective optimization is employed to address the trade-off between pipeline resource utilization and transportation time. The core challenge lies in simultaneously achieving the following two objectives:  
* Minimization of the total cost based on pipeline node weights: The weight of a pipeline node represents its associated resource demand and operational complexity. Selecting nodes with lower weights helps reduce overall task costs and improves scheduling efficiency;
* Minimization of edible oil transportation time: Reducing the transportation time directly enhances system throughput, which is particularly critical in handling urgent or high-priority tasks.  
In the context of multi-objective optimization, a Pareto-optimal analysis based on pipeline node weight cost and transportation time is adopted to guide task prioritization and dynamic scheduling. The process consists of the following steps:  
1. Pareto Front Analysis: Task cost (based on node weights) and transportation time are computed and displayed on a Pareto scatter plot. Points closer to the Pareto front indicate superior performance in both cost and time.
2. Multi-Objective Ranking and Decision Mechanism: When the cost difference between the top-ranked task and the rest is below a predefined threshold \(w_{cost}\), the system further evaluates whether the top task also has the shortest transportation time. 1. If the highest-ranked task does not have the shortest transportation time, the system reassigns priority to the task with the minimum delivery time and updates the scheduling order accordingly; 2. This decision-making process jointly considers pipeline cost and transportation efficiency, ensuring a balanced trade-off between operational performance and resource utilization.  
The integration of multi-objective optimization theory into real-world edible oil pipeline scheduling introduces a dynamic and scientifically grounded scheduling framework. This approach addresses the shortcomings of traditional single-objective strategies, which often lead to inefficiencies and resource underutilization. It is particularly effective in complex factory environments characterized by variable task requirements and high-priority constraints.  
Figure 6: Pareto distribution diagram.