FH Brewery consists of three different locations $($Wichita$,$ Kansas City and Tulsa$)$ brewing three different kinds of beers $($Golden Sunshine $($ale$),$ Cow Pie $($milk stout$),$ and I$49($wheat$)).$ Note: Wichita does not brew I$49,$ and KC does not brew Golden Sunshine.

For the upcoming production period, we want to optimize $($max profits $$ sales less costs$)$ production of the three types of beers at the three locations. The typical brewing scenario has been simplified for the sake of this exercise.

There are two types of resources we will model in beer production $$ materials and processing time. The amount of each resource used per barrel of beer produced is shown in the table below.

For instance, to make a barrel of Golden Sunshine $($anywhere$),3$ units of material are required and $1\mathrm{.}5$ processing time units are needed. Each barrel of Golden Sunshine made $($regardless of location$)$ provides sales of $$94.$ Similarly, the data is shown for the other two products.

Costs are calculated based upon the location of the brewery $($see table$)$ and the material and processing time required to make each beer. For instance, each unit of material used to brew a barrel of beer at Wichita costs $$8$ per unit, and each unit of processing time costs $$21$ per unit. This is independent of beer type $($i$.$e$,$ same cost regardless of beer $$ each beer simply uses different amounts$).$ The table shows similar information for the other two locations. $($ An example of how to calculate this cost and thus the profit is shown at the end$).$

Available brewing materials are treated at an overall company level $$ they can be allocated in any amounts to the three locations. Consider having a total of $1285$ units of materials available to brew beer for the next period.

Processing time is location dependent and cannot be $$shared$$ between locations. Wichita has $220$ processing time units available, Kansas City has $175$ units available, and Tulsa $240$ units available for the next period.

Product mix requirements involve several constraints. Overall, at least $105$ barrels of each beer type must be produced. Relatedly, any one beer type cannot exceed $40\%$ of overall total beer production.

Quality of beer varies by the combination of location and beer type, as also shown in the table above. For instance, the quality of Golden Sunshine beer made in Wichita is $5\mathrm{.}0,$ whereas the quality of Cow Pie at Wichita is $4\mathrm{.}4.$ The average overall quality of Golden Sunshine and Cow Pie must be at least $4\mathrm{.}5($treated individually$),$ while the average overall quality of I$49$ must be at least $4\mathrm{.}25.$ Note that if there is not a quality table entry for the beer$/$location combination, you cannot produce that beer at that location.

Create an LP model that will determine the optimal way to brew the three beers across the three locations. Maximize the sales less costs $($profit$).$ Include all relevant constraints. After creating and validating your model, add integer constraints to ensure whole numbers of barrels are produced.

This is version $1\mathrm{.}0$ of the problem. I reserve the right to fix errors and other aspects of this problem as they become known up until the due date of the checkpoint.

Calculating Profit: Sales less Cost.

Example: Golden Sunshine for Wichita.

Uses $3$ material units @ $$8/$unit$.3$ x $$8=$ $$24$ material cost.

Uses $1\mathrm{.}5$ process units @ $$21/$unit $1\mathrm{.}5$ x $$21=$ $$31\mathrm{.}5$ process cost.

Sales: $$94$ unit.

Profit $=$ Sales less costs: $942431\mathrm{.}5=$ $$38\mathrm{.}5$ per barrel of Golden Sunshine for Wichita.