## 4 Multi-Objective Optimization for Edible Oil Transportation Scheduling Method
### Edible Oil Transportation Time Function  
In the production process of edible oil plants, transportation time is a critical factor for scheduling and operational efficiency. This study estimates the transportation time of various edible oils based on the Hagen-Poiseuille law, which describes fluid flow through pipelines, and utilizes these estimations to arrange production sequences accordingly. The physical properties of edible oils, especially their viscosity coefficients, significantly affect their transit times within the pipelines. According to [30], the viscosity and temperature characteristics vary among different types of edible oils, impacting flow behavior. Edible oils with higher viscosity require longer transportation times due to increased flow resistance, whereas those with lower viscosity can be transported faster under the same pipeline conditions. These differences impact not only the efficiency of individual transportation tasks but the overall scheduling and planning of production, especially in multi-product, multi-batch manufacturing scenarios. By accounting for the viscosity characteristics of edible oils in transport scheduling, the system effectively avoids prolonged transit times and pipeline congestion, thereby enhancing production efficiency and reducing operational costs [30; 31; 32]. This study is conducted in a region characterized by a subtropical monsoon and tropical monsoon climate, with ambient temperatures fluctuating between approximately 15 \({}^{\circ}\)C and 35 \({}^{\circ}\)C. Accordingly, the viscosity coefficients of edible oils are standardized at an average temperature of around 25 \({}^{\circ}\)C, referencing 23.9 \({}^{\circ}\)C as reported in [30]. Due to variations in oil density, conversions between mass and volume are required for relevant calculations. To ensure data consistency, this study adopts the density conversion values provided in [33]. Based on the specified viscosity and density parameters, oil transport time is then calculated using fluid dynamics principles. Temperature also affects the viscosity coefficient of the oil. As temperature increases, the viscosity coefficient of edible oil typically decreases, resulting in increased fluidity; conversely, as temperature decreases, the viscosity coefficient increases, leading to reduced fluidity. According to the analysis presented in [34], this study establishes a table correlating the viscosity coefficient of edible oil with temperature to enhance the accuracy of time function calculations. Specifically, the Hagen-Poiseuille law serves as the governing model to estimate the edible oil flow velocity v(m/s), incorporating pressure differential, pipe radius, viscosity coefficient, and pipe length as key variables [35; 36]. The governing equation is defined as follows:  
\[v=\frac{\Delta P\times r^{2}}{8\times\eta\times L^{\prime}}, \tag{7}\]  
Here, \(\Delta P\) denotes the pressure difference between the two ends (Pa), \(r\) is the pipeline radius (m), \(\eta\) represents the viscosity coefficient of the edible oil (Pa-s), and \(L\) is the total pipeline length required for the transportation task (m). This equation indicates that, as the viscosity coefficient increases, the flow velocity significantly decreases, thereby extending the transportation time. This physical property is a critical factor in edible oil transportation, since the viscosity can vary considerably among different oils. Consequently, differences in transportation time must be accounted for and adjusted within production scheduling. The actual transportation time of the edible oil can be further estimated using the volumetric flow rate formula, as follows:  
\[Q_{v}=A\times v, \tag{8}\]  
\[Q_{v}=V/t, \tag{9}\]  
where \(Q_{v}\) represents the volumetric flow rate (m\({}^{3}\)/s), \(A\) denotes the cross-sectional area of the pipeline (m\({}^{2}\)), \(V\) is the volume of the fluid (m\({}^{3}\)), and \(t\) is the time (min).