Formulate the following problems as linear optimization models. Where appro$-$ priate, it may be useful to number the commodities and then use subscripted vari$-$ ables. In some cases no specific objective is named; it is up to the reader to come up with an appropriate objective.

Each problem could be interpreted as having three parts:

$($a$)$ Define the variables and give the algebraic model.

$($b$)$ Make a model in LINGO or the Excel Solver and find the solution. $($c$)$ Give a recommendation in words to management.