You assist a client to select a location $x=(x_1,x_2)$ for a new service facility that will serve $K=50$ customers

by providing a service to each customer $($such as visiting the customer a given number of times per year$).$

We assume that the new facility can be located anywhere within the unit square $0x_{1}1,0x_{2}1$ : this

square models a $100$ km by $100$ km rectangular region. Customer locations are modeled by given points

$p_k=(p_{k1},p_{k2})$ for $k=1,$dots,$K$ located within the unit square. Each customer's yearly demand for the service

is assumed to be known; we also assume that all demands must be satisfied. Customers can have different

yearly demands for service. This aspect is expressed by assigning weight $w_k$ to customer $k=1,$dots,$K.$ To

illustrate the problem, please see the figure below that shows the unit square $($blue$),$ a possible but not

optimized location for the facility $($black dot$),$ and the locations of the weighted customers $($red dots of

radius $w_k)$ for $k=1,$dots,$K.$

The distance between the facility location $x=(x_1,x_2)$ and customer $k$ located at point $p_k=(p_{k1},p_{k2})$ is

expressed by the Euclidean $(1_2-$norm$)$ distance function defined as

$d(x,p_k)=\sqrt[\phantom{2}]{(x_1-p_{k1}{)}^2+(x_2-p_{k2}{)}^2}$

This model corresponds to reaching the facility on the straight line segment connecting it to the customer.

$($An example of such a scenario is serving customers by helicopter flights.$)$

Formulate a decision model that optimizes the location of the facility. The quality of a location is modeled

by the weighted sum of all distances between the facility and the customers.

At your discretion, YOU can find a lot of professional literature on this important type of problem.