# Globally Optimal Facility Locations for Continuous-Space Facility Location Problems
## 3 Location Estimation Based on the Manhattan Distance
### Minimum Distance Approach  
In this subsection, we provide the three-dimensional single-facility location methods for the Manhattan distance. With the objective function of \(\Psi_{3}\) in (3), let \((\vec{u},\vec{v},\vec{w})\) be the optimal facility location. Then, we have  
\[(\vec{u},\vec{v},\vec{w})=\underset{(u,v,w)}{\text{argmin}}(\Psi_{3})= \underset{(u,v,w)}{\text{argmin}}\Big{[}\sum_{i=1}^{n}\omega_{i}(|x_{i}-u|+|y_ {i}-v|+|z_{i}-w|)\Big{]} \tag{4}\]  
The minimizer of \(\Psi_{3}\) in (3), denoted by \((\vec{u},\vec{v},\vec{w})\), can be obtained by the following estimating equations that need to be solved for \(u\), \(v\), and \(w\) according to Section 1.3 of Hettmansperger and McKean [57]. \[\frac{\partial\text{Obj}_{3}}{\partial u}=\sum_{i=1}^{n}\omega_{i}\text{sgn }(x_{i}-u)=0 \tag{5}\]  
\[\frac{\partial\text{Obj}_{3}}{\partial v}=\sum_{i=1}^{n}\omega_{i}\text{sgn }(y_{i}-v)=0 \tag{6}\]  
\[\frac{\partial\text{Obj}_{3}}{\partial w}=\sum_{i=1}^{n}\omega_{i}\text{sgn }(z_{i}-w)=0 \tag{7}\]  
Then, the optimal values \(\vec{u}\), \(\vec{v}\), and \(\vec{w}\) can be calculated separately. It is easily seen that they are the weighted medians of the \(x\)-axis, \(y\)-axis, and \(z\)-axis observations, which will be detailed later. Note that the weighted median was first suggested by [PERSON] [58] and since then it has been widely used in many applications [59]. As an illustration, we briefly introduce the conventional median first and then the weighted median. As the values \(\vec{u}\), \(\vec{v}\), and \(\vec{w}\) can be obtained separately, we only consider the weighted median for the \(x\)-axis observations. Then, the weighted medians for the \(y\)-axis and \(z\)-axis observations are easily obtained using the same method. **Definition 1**. : _Given a set of observations \(x_{1}\),\(x_{2}\),..., \(x_{n}\), the empirical cumulative distribution function \(F_{n}\) is defined as_  
\[F_{n}(x)=\frac{1}{n}\sum_{i=1}^{n}\text{I}(x_{i}\leq x),\ x\in\mathbb{R} \tag{8}\]where \(\mathrm{I}(A)\) represents the indicator function defined as  
\[\mathrm{I}(A)= \begin{cases}1&\text{if $A$ is satisfied}\\ 0&\text{otherwise}\end{cases}  \tag{9}\]  
**Definition 2**. : [PERSON] [60] defined the left sample quantile function (inverse cumulative distribution function) as below._  
\[F_{n,L}{}^{-1}(p)=\inf\left\{x:F_{n}(x)\geq p\right\} \tag{10}\]  
Using this, the conventional median \(F_{n}{}^{-1}(1/2)\) is obtained as:  
\[F_{n,L}{}^{-1}\bigg{(}\frac{1}{2}\bigg{)}=\inf\left\{x:F_{n}(x)\geq\frac{1}{2} \right\}=x_{(k)}\text{ if }\frac{k-1}{n}<\frac{1}{2}\leq\frac{k}{n},\ k=1,2, \ldots,n \tag{11}\]  
where \(x_{(1)}\leq x_{(2)}\leq\ldots\leq x_{(n)}\). **Definition 3**. : _[PERSON] [61]defined the right sample quantile function (inverse cumulative distribution function), which is given by_  
\[F_{n,R}{}^{-1}(p)=\inf\left\{x:F_{n}(x)>p\right\} \tag{12}\]  
Note that the definition by [PERSON] [61] is slightly different to that of [PERSON] [61]. It is easily seen that  
\[\inf\left\{x:F_{n}(x)\geq p\right\}\leq\inf\left\{x:F_{n}(x)>p\right\} \tag{13}\]  
Thus, the sample quantile by [PERSON] [60] is called the left quantile, while the sample quantile by [PERSON] [61] is the right quantile. [PERSON] [62] and [PERSON] [63] showed that  
\[F_{n,R}{}^{-1}(p)=\inf\left\{x:F_{n}(x)>p\right\}=\sup\left\{x:F_{n}(x)\leq p\right\} \tag{14}\]  
Based on the quantile function by [PERSON] [61], we have the corresponding median \(F_{n}{}^{-1}(1/2)\), which is obtained as:  
\[F_{n,R}{}^{-1}\bigg{(}\frac{1}{2}\bigg{)}=\inf\left\{x:F_{n}(x)>\frac{1}{2} \right\}=x_{(k+1)}\text{ if }\frac{k-1}{n}<\frac{1}{2}\leq\frac{k}{n},\ k=1,2, \ldots,n \tag{15}\]  
where \(x_{(1)}\leq x_{(2)}\leq\ldots\leq x_{(n)}\). It is obvious that the difference between (13) and (15) is the case when \(F_{n}(1/2)=0.5\) with even \(n\). It is obvious that (13) takes the left bound value and (15) takes the right bound value. Moreover, both of them are the minimizers (medians) to the total Manhattan distance of the \(x\)-axis observations.