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.

$\backslash$begin$\{$tabular$\}\{|$l$|$c$|$c$|$c$|\}$

$\backslash$hline $\backslash$multicolumn$\{1\}\{|$c$|\}\{$ CITY $\}$ & $\backslash$begin$\{$tabular$\}\{$c$\}$

DISTANCE EAST OF $\backslash \backslash$

DURBAN

$\backslash$end$\{$tabular$\}$ & $\backslash$begin$\{$tabular$\}\{$c$\}$

DISTANCE NORTH OF $\backslash \backslash$

DURBAN

$\backslash$end$\{$tabular$\}$ & PRODUCT TONNAGE $\backslash \backslash$

$\backslash$hline Umhlaba'uyalingana $($A$)$ & $90$ & $10$ & $240,000\backslash \backslash$

$\backslash$hline Dumbe $($B$)$ & $140$ & $60$ & $320,000\backslash \backslash$

$\backslash$hline Mbembeswane $($C$)$ & $20$ & $70$ & $450,000\backslash \backslash$

$\backslash$hline Maqonqo $($D$)$ & $50$ & $20$ & $120,000\backslash \backslash$

$\backslash$hline Msunduzi $($E$)$ & $100$ & $120$ & $60,000\backslash \backslash$

$\backslash$hline Ulundi $($F$)$ & $10$ & $140,000\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}$

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

$\backslash ((12\backslash )$

location for this new sugar mill.

marks$)$

All distance references use the city of Durban $($Headquarters$)$ as $\backslash ((0,0)\backslash ).4\mathrm{.}2$ 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.