Agile Logistics

The company Agile Logistics is considering where to construct a new production factory to supply shipments to six cities. All shipments are direct, i$.$e$.,$ a truck travels from the factory to one city only and returns back to the factory. The city locations as well as the required number of shipments to each city per year are given in the table below.

City location $($X$,$Y$)($km$,$ km$)$

City X Y Number of shipments

$1232030$

$253250$

$3123119$

$4243522$

$582541$

$6151020$

Assume any location for the new factory is feasible and that the distance that shipments have to travel from the factory to each city can be estimated by the Euclidean distance: $(($x$\_1-$x$\_2)^2+($y$\_1-$y$\_2)^2).$ The company pays $$10$ for every km a shipment travels.

Formulate an NLP model to determine where the new factory should be located such that the total yearly shipping cost is minimized.

Suppose the objective is to minimize the maximum distance of the factory from any city. Modify your model accordingly and resolve the problem. Notice what happens to the original objective value when you optimize the new objective value.