Question $1(60$ points$)$

Q$1\mathrm{.}1.(20$ points$)$

You assist your client to select a location $x=(x_1,x_2)$ for a service facility that will serve $K=20$ customers

by providing a single $($identical$)$ service or commodity to each customer. The facility can be located

anywhere within the unit square $0x_{1}1,0x_{2}1$; this square can model e$.$g$.,$ a $100km$ by $100km$

rectangular region scaled to the unit square. The customers are modeled by points $p_k=(p_{k1},p_{k2})$ for $k=1,$

$...,K$ located within the unit square. Each customer's yearly demand for the service is assumed to be a

known value: we also assume that all demands must be satisfied. Customers can have different relative

weights, proportionate to the quantity of their yearly demand. This aspect is expressed by assigning weights

$w_k$ for $k=1,$dots,$K$ to the customers. Assume that the distance between the facility location $x$ and customer

$k($point $p{k}_k)$ is expressed by the Euclidean $(12-$norm$)$ distance function defined by

$d(x,p{k}_k)=\sqrt[\phantom{2}]{(x_1-p_{k1}{)}^2+(x_2-p{k}_2{)}^2}.$

Assume additionally that the regions $2x_1-3x_2>1,4x_1+3x_2>5,x_1^2-5x_2^2>0,$ and $3x_1^2+5x_2^2<mn\; id="EditorElement\_756088">2</mn>$

must be excluded from consideration for the possible location of the facility.

Refer to Assignment $1$ of the semester for further details as needed.

Formulate the model.

Q$1\mathrm{.}2.(10$ points$)$

Determine the convexity properties of the stated facility location model.

Q$1\mathrm{.}3.(30$ points$)$

To develop a numerical example in Excel, choose locations for the customers within the region $0x_{1}1,$

$0x_{2}1.$ Each students should create a different numerical example. Solve your model. Generate an

Answer Report, and display the customer and facility configuration on a figure in your Excel file.