set up the linear programming problem. Do not solve. A brewery manufactures three types of beer-lite, regular and dark. Each vat of lite beer requires 6 bags of barley, 1 bag of sugar, and 1 bag of hops. Each vat of regular bear requires 4 bags of barley, 3 bags of sugar, and 1 bag of hops. each vat of dark beeer requires 2 bags of barley, 2 bags of sugar, and 4 bags of hops. each day the brewery realizes a profit of $10 per vat of lite beer, $20 per vat of regular beer, and $30 per vat of dark beer. how many vats of lite, regular and dark beer should be brewed in order to maximize profits?
a) write down the linear programming problem, including the objective function and system of linear inequalities that constrain the problem, defining all your variables.
To set up the linear programming problem, let's define the following variables:
Let L = number of vats of lite beer to produce
Let R = number of vats of regular beer to produce
Let D = number of vats of dark beer to produce
The objective is to maximize the profit, which can be expressed as:
Profit = 10L + 20R + 30D
Now, let's consider the constraints:
1) Barley Constraint:
Each vat of lite beer requires 6 bags of barley, each vat of regular beer requires 4 bags of barley, and each vat of dark beer requires 2 bags of barley.
So the total amount of barley needed is given by:
6L + 4R + 2D ≤ B, where B represents the total available bags of barley.
2) Sugar Constraint:
Each vat of lite beer requires 1 bag of sugar, each vat of regular beer requires 3 bags of sugar, and each vat of dark beer requires 2 bags of sugar.
So the total amount of sugar needed is given by:
1L + 3R + 2D ≤ S, where S represents the total available bags of sugar.
3) Hops Constraint:
Each vat of lite beer requires 1 bag of hops, each vat of regular beer requires 1 bag of hops, and each vat of dark beer requires 4 bags of hops.
So the total amount of hops needed is given by:
1L + 1R + 4D ≤ H, where H represents the total available bags of hops.
4) Non-negativity Constraints:
L ≥ 0, R ≥ 0, D ≥ 0
Therefore, the linear programming problem can be written as:
Maximize: Profit = 10L + 20R + 30D
Subject to:
6L + 4R + 2D ≤ B
1L + 3R + 2D ≤ S
1L + 1R + 4D ≤ H
L ≥ 0, R ≥ 0, D ≥ 0
Note: B, S, and H represent the total available bags of barley, sugar, and hops, respectively.
To set up the linear programming problem, we need to define our variables, formulate the objective function, and set up the system of linear inequalities that represent the constraints.
Let's begin by defining our variables:
Let L be the number of vats of lite beer to be brewed.
Let R be the number of vats of regular beer to be brewed.
Let D be the number of vats of dark beer to be brewed.
Now, let's define the objective function. The objective is to maximize profits, which can be achieved by maximizing the total profit earned from brewing and selling the different types of beer.
The profit earned from each vat of lite beer is $10, so the total profit from lite beer is 10L.
The profit earned from each vat of regular beer is $20, so the total profit from regular beer is 20R.
The profit earned from each vat of dark beer is $30, so the total profit from dark beer is 30D.
Therefore, the objective function is:
Z = 10L + 20R + 30D
Next, let's set up the system of linear inequalities based on the given constraints:
1. Barley constraint:
Each vat of lite beer requires 6 bags of barley, each vat of regular beer requires 4 bags of barley, and each vat of dark beer requires 2 bags of barley.
The total supply of barley available is limited, let's say it is B bags.
Therefore, the constraint for barley is:
6L + 4R + 2D ≤ B
2. Sugar constraint:
Each vat of lite beer requires 1 bag of sugar, each vat of regular beer requires 3 bags of sugar, and each vat of dark beer requires 2 bags of sugar.
The total supply of sugar available is limited, let's say it is S bags.
Therefore, the constraint for sugar is:
1L + 3R + 2D ≤ S
3. Hops constraint:
Each vat of lite beer requires 1 bag of hops, each vat of regular beer requires 1 bag of hops, and each vat of dark beer requires 4 bags of hops.
The total supply of hops available is limited, let's say it is H bags.
Therefore, the constraint for hops is:
1L + 1R + 4D ≤ H
4. Non-negativity constraint:
Since the number of vats cannot be negative, we have:
L ≥ 0
R ≥ 0
D ≥ 0
These are the linear programming problem's objective function and the system of linear inequalities that constrain the problem, with all the variables defined.