# Vehicle Route Planning for Relief Item Distribution under Flood Uncertainty
## 3 The Proposed Model
### Population Sizing Module  
The allocation of the size of the enhanced population has an impact on the quality of the solutions and the processing speed. The allocation proportions are 25%, 50%, 75%, and 100% of the total population. It is found that setting the allocation proportions at 25% and 75% yields good solution quality for problems where node distributions are evenly spread and clustered [31], respectively. However, because victim location data are given in coordinate positions, the actual distribution pattern of the nodes is unknown, making it difficult to determine an appropriate enhanced population size. This section presents a  
Figure 5: An example of solution and population representation.  
technique for selecting an appropriate population size by analyzing the node distribution pattern. This module is referred to as the Population Sizing Module (PSM). In the first step, a grid of suitable size for node distribution is generated. The second step uses the standard deviation (SD) to analyze the distribution pattern. This article assumes that a uniform distribution means that within each grid cell, there is one node, and the standard deviation value is 0. Similarly, if the SD > 1.0, it indicates that the nodes are clustered, meaning that more than 68.2% of the nodes are close to each other, as illustrated in Algorithms 1-3.  
```
0: List of nodes \(xy\) with their \((x,y)\) coordinates
0: A grid with each cell containing the count of nodes assigned to it \(minX,minY,maxX,maxY\leftarrow\min(xy),\max(xy)\) \(width\gets maxX-minX\) \(height\gets maxY-minY\) \(n,m\leftarrow\text{OptimizeGridDimensions(length of $xy$, width, height)}\)\(\triangleright\) Call Algorithm 2. \(cellWidth\gets width/n\) \(cellHeight\gets height/m\) \(grid\leftarrow\text{Initialize an $m\times n$ matrix with all values set to 0}\) for each node in \(xy\)do \(xIndex\leftarrow\lfloor(node\_x-minX)/cellWidth\rfloor\) \(yIndex\leftarrow\lfloor(node\_y-minY)/cellHeight\rfloor\) \(grid[yIndex][xIndex]\gets grid[yIndex][xIndex]+1\) endfor  Return grid
```  
**Algorithm 1** Generate Spatial Grid.  
```
0: Total number of nodes (nodes), area width (\(w\)), and height (\(h\))
0: Optimized grid dimensions (grid_w, grid_h) that best fit the area's aspect ratio ratio \(\gets w/h\) \(grid\_w\leftarrow\lceil\sqrt{nodes}\rceil\) \(grid\_h\leftarrow\lceil nodes/grid\_w\rceil\) bestDiff \(\leftarrow\infty\) bestDimensions \(\leftarrow(grid\_w,grid\_h)\) while true do \(newGridW\gets grid\_w+1\) \(newGrid\leftarrow\lceil nodes/newGridW\rceil\) \(newRatio\leftarrow newGridW/newGridH\) \(ratioDiff\leftarrow\lfloor newRatio-ratio\rfloor\) if\(ratioDiff<bestDiff\)then \(bestDiff\gets ratioDiff\) \(bestDimensions\leftarrow(newGridW,newGridH)\) \(grid\_w\leftarrow newGridW\) else break endif endwhile if\(h>w\)then return\((bestDimensions[1],bestDimensions[0])\) else return\(bestDimensions\) endif
```  
**Algorithm 2** Optimize Grid Dimensions.  
```
0: Total number of nodes (\(nodes\)), area width (\(w\)), and height (\(h\))
0: Statistical analysis of the node distribution within the grid \(diff\_grid\leftarrow\) (Number of rows in grid \(*\) Number of columns in grid) \(-\) Length of \(xy\) \(count\gets 0\) for each row in grid do while\(count<diff\_grid\) and there is a zero in the row do  Remove a zero from the row  Increment count by 1 endwhile endfor \(sd\leftarrow\) Calculate standard deviation from all cell counts in the grid
```  
**Algorithm 3** Analyze Distributions.