QUESTION $4$

$(25$ marks$)$

Karoo $($Pty$)$ Ltd sells $500ml$ bottles of water to various restaurant groups in South Africa..

Karoo collects the water from a local dam, purifies the water by filtering the water through different processes and treating it$.$ Three types of bottled water are produced namely Still, Sparkling and Flavoured.

The selling and manufacturing cost of the respective types of water is as follow:

$\backslash$table$[[,$Still,Sparkling,Flavoured$],[$Selling price,R$10,$R$12,$R$15],[$Manufacturing cost:$,,,],[-$material,R$1,$R$2,$R$3],[-$labour,R$1,$R$1,$R$3],[-$variable overhead,R$1,$R$2\mathrm{.}50,$R$2],[-$fixed overhead,R$3,$R$5,$R$4],[$Profit$,$R$4,$R$1\mathrm{.}50,$R$3]]$

Due to the continued drought in the Cape region, demand for Still bottled water has increased substantially. Consequently, Karoo is anticipating operating constraints soon. Management are concerned that the demand is of a temporary nature and they are therefore not prepared to invest in expanding manufacturing capacity at this time.

Karoo has $1000$ direct labour hours and $4000$ machine hours available. Each $500ml$ bottle of water requires the following time to bottle:

$\backslash$table$[[$Direct labour hours,$0\mathrm{.}5,0\mathrm{.}1,0\mathrm{.}1],[$Machine hours,$0\mathrm{.}1,0\mathrm{.}3,0\mathrm{.}2]]$

Due to the uncertainty of rainfall in the Cape region, the following outcomes can be assumed:

Alternative $1$: It rains

If it rains there will be a smaller demand for Still Water. Based on prior years' consumption, Karoo expects orders totaling $1500$ bottles of Still, $8000$ bottles Sparkling and $6000$ bottles of Flavoured water.

Alternative $2$: Drought

Karoo anticipates a temporary increase of in demand of Still water bottles. The demand for Still water during the next $2$ months will be $20000$ bottles. Sparkling is expected to remain at its current level of $8000$ bottles and no Flavoured water will not be bottled during this period.

$14$

REQUIRED:

$4\mathrm{.}1$ Recommend the optimal production mix and contribution for Alternative

$(6)$

$1$: It rains.

$4\mathrm{.}2$ Recommend the optimal production mix for alternative $2$: Drought that

$(10)$

will maximise profit.

$4\mathrm{.}3$ Discuss any qualitative factors that management should consider if alternative $2$ is implemented..

$4\mathrm{.}4$ Explain the assumptions underlying the linear programming model.

$(4)$