# Globally Optimal Facility Locations for Continuous-Space Facility Location Problems
## 3 Location Estimation Based on the Manhattan Distance
### Objective Function Based on the Manhattan Distance  
The Manhattan distance is usually applied in the urban area of interest due to the street configuration [40; 56]. For instance, the determination of the bicycle-sharing stations or the ambulance stations needs to consider the Manhattan distance. For more details on the Manhattan distance, please refer to [PERSON] and [PERSON] [7]. Based on the Manhattan distance, the general objective function (minimization of the total weighted distance) can be formulated with the following notations.  
Parameters: \(n\), Number of demand points, \((i=1,2,\ldots,n)\); \(x_{i}\), \(x\)-coordinate of demand point \(i\); \(y_{i}\), \(y\)-coordinate of demand point \(i\); \(z_{i}\), \(z\)-coordinate of demand point \(i\); \(w_{i}\), Weight (Priority level) of demand point \(i\); Decision variables: \(u\), \(x\)-coordinate of the facility; \(v\), \(y\)-coordinate of the facility; \(w\), \(z\)-coordinate of the facility;  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Article** & **Distance Type** & **Method** & **Optimal Estimator** \\ \hline [PERSON] [44] & Euclidean distance & CG method & No \\ [PERSON] and [PERSON] [50] & Squared Euclidean distance & Clustering algorithm & Yes \\ [PERSON] and [PERSON] [51] & Euclidean distance & Clustering algorithm & No \\ [PERSON] and Buyiksaqt [33] & Euclidean distance & Clustering algorithm & No \\ [PERSON] and [PERSON] [37] & Euclidean distance & Clustering algorithm & No \\ [PERSON] and [PERSON] [52] & Euclidean distance & Select from potential locations & No \\ [PERSON], [PERSON] [45] & Euclidean distance & Clustering algorithm & No \\ [PERSON], [PERSON] [38] & Euclidean distance & Clustering algorithm & No \\ [PERSON] [46] & Manhattan distance and Euclidean & CG method & No \\  & distance & Clustering algorithm & No \\ [PERSON] and [PERSON] [39] & Euclidean distance & Clustering algorithm & No \\ [PERSON], [PERSON] [53] & Euclidean distance & Clustering algorithm & No \\ [PERSON] and [PERSON] [54] & Manhattan distance & Clustering algorithm & No \\ [PERSON], [PERSON] [22] & Manhattan distance & Bi-level decomposition & No \\  & & algorithm & No \\ [PERSON], [PERSON] [55] & Euclidean distance & Artificial Bee Colony-based & No \\ [PERSON], [PERSON] [40] & Euclidean distance & Clustering algorithm & No \\ This paper & Manhattan distance & Exact method & Yes \\ \hline \hline \end{tabular}
\end{table}
Table 1: Summary of the literature about the continuous-space location problems.  
Generally, the objective function needs to be minimized so that the total weighted distance-related travel time or cost is optimized. Before the objectives are presented, the weights need to be normalized using the following equation.  
\[\omega_{i}=\frac{w_{i}}{\sum_{i=1}^{n}w_{i}} \tag{1}\]  
where \(\omega_{i}\) is the normalized weight of demand point \(i\).  
Then, the objective function for the two-dimensional single-facility location problem is given by  
\[\Psi_{2}=\sum_{i=1}^{n}\omega_{i}(|x_{i}-u|+|y_{i}-v|) \tag{2}\]  
where \(|x_{i}-u|+|y_{i}-v|\) is the distance measured along the axis at right angles. Please note that the normalized weight influences the objective function value, but it would not affect its final solution. Similarly, the same situation happens in the other objective functions.  
The objective function for the three-dimensional single-facility location problem is given by  
\[\Psi_{3}=\sum_{i=1}^{n}\omega_{i}(|x_{i}-u|+|y_{i}-v|+|z_{i}-w|) \tag{3}\]  
where \(|x_{i}-u|+|y_{i}-v|+|z_{i}-w|\) is the distance measured along the axis at right angles, which is usually applied in the manison building of interest and three-dimensional facility cases.