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 KwaZulu$-$

Natal 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.

$4\mathrm{.}1$ Using the provided information, use the center$-$of$-$gravity method to calculate the coordinates of the best

location for this new sugar mill.

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

$4\mathrm{.}2$ SweetHarvest's operations include the cultivation of sugarcane, its transportation to processing plants, and the

conversion of raw cane into refined sugar and other products. It is considering opening a retail shop for its sugar products

and currently has three alternative locations. The fixed and variable costs associated with each of the alternatives are

provided below,

$4\mathrm{.}2\mathrm{.}1$ Assuming revenue is the same regardless of the location selected. Identify the range over which each

location is giving the best outcomes.