UB Food Products has three processing plants $($A$,$ B and C$),$ where different variables of potato products are produced. Recently, the company has experienced a growth in its product demand, so it wants to buy potatoes from new farms. The company is considering six new farms with the following projected harvest:

Table $4-1$

Farm $123456$

Projected Annual Harvest

$($thousands of tons$)11\mathrm{.}210\mathrm{.}512\mathrm{.}89\mathrm{.}310\mathrm{.}89\mathrm{.}6$

The company currently has the following additional available production capacity $($tons$)$ at its three plants, which it wants to utilise:

Table $4-2$

Plant A B C

Available Capacity

$($thousands of tons$)121014$

The shipping cost per ton from the farms being considered for purchase to the plants are as follows:

Table $4-3$

Farm Plant $($Shipping costs, $/$ton$)$

A B C

$1181512$

$2131017$

$3161418$

$4191516$

$5171912$

$6141612$

$($a$)$ Formulate a mathematical model that can be used to find farms that will meet available production capacity at a minimum cost. Solve it$.(6$ points$)$

$($b$)$ With the increase in the price of potatoes, the management team plan to purchase farms instead of purchasing potatoes from them. The annual fixed cost of each farm is summarised in Table $4-4.$ Build a model to help the manager to select the farms that will meet available production capacity at a minimum cost. Solve it$.(6$ points$)$

Table $4-4$

Farm $123456$

Fixed Annual Costs

$(1,000$s$)405390450368520465$

$($c$)$ After reviewing the solution to part $($b$),$ the manager realised that additional requirements must be specified. If farm $2$ is purchased, then farm $6$ cannot be purchased but farm $4$ must be purchased. Revise your model to incorporate these additional requirements and solve for the optimal solution. $(6$ points$)$

$($d$)$ Comment $($discussion or analysis beyond$/$based$)$ on the solutions of $($a$-$c$).(7$ points$)$