QUESTION $4$

$(20$ Marks$)$

SweetHarvest Pty Ltd is a leading sugar processing company based in South Africa. Established in the early $1990$s$,$ SweetHarvest has grown to become a significant player in the agricultural sector, specializing in the production of refined sugar, molasses, and other by$-$products. The company operates several large$-$scale processing plants across the KwaZuluNatal and Mpumalanga regions, sourcing sugarcane from local farmers and its own plantations. The organisation plans to build a new sugar mill in Southwestern KZN$.$ The primary objective of the mill is to provide the farmers with a place to take their crop for processing that will reduce their transportation costs. The operations teams for the organization believes that the center$-$of$-$gravity method is appropriate for this objective. There are approximately $200$ suppliers across this region who are clustered within six cities. The following data has been provided as part of the analysis.

$\backslash$table$[[$CITY$,\backslash$table$[[$DISTANCE EAST OF$],[$DURBAN$]],\backslash$table$[[$DISTANCE NORTH OF$],[$DURBAN$]],$PRODUCT TONNAGE$],[$Umhlaba$\text{'}$uyalingana $($A$),90,10,240,000],[$Dumbe $($B$),140,60,320,000],[$Mbembeswane $($C$),20,70,450,000],[$Maqonqo $($D$),50,20,120,000],[$Msunduzi $($E$),100,120,140,000],[$Ulundi $($F$),10,8000,]]$

$4\mathrm{.}1$ Using the provided information, use the center$-$of$-$gravity method to calculate the coordinates of the best Location for this new sugar mill.

$(12$ marks$)$

All distance references use the city of Durban $($Headquarters$)$ as $(0,0).$