12) Betty Malloy, owner of the Eagle Tavern in Pittsburgh, is preparing for Super Bowl Sunday, and she must determine how much beer to stock. Betty stocks three brands of beer—Yodel, Shotz, and Rainwater. The cost per gallon (to the tavern owner) of each brand is as follows: Brand Cost/Gallon Yodel $1.50 Shotz 0.90 Rainwater 0.50 The tavern has a budget of $2,000 for beer for Super Bowl Sunday. Betty sells Yodel at a rate of $3.00 per gallon, Shotz at $2.50 per gallon, and Rainwater at $1.75 per gallon. Based on past football games, Betty has determined the maximum customer demand to be 400 gallons of Yodel, 500 gal- lons of Shotz, and 300 gallons of Rainwater. The tavern has the capacity to stock 1,000 gallons of beer; Betty wants to stock up completely. Betty wants to determine the number of gallons of each brand of beer to order so as to maximize profit.
a. Formulate a linear programming model for this problem.
X gallons of yodel
Y gallons of shotz
Z gallons of rainwater
Cost
Price
Yodel
$1.50
$3.00
Shotz
$.90
$2.50
Rainwater
$.50
$1.75
P=(3.00x+2.50y+1.75z)- (1.50z+.90y+.50z)
(Max)P=1.50x+1.60y+1.25z
P= income-cost
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you need to write your constraint equations.
To formulate a linear programming model for this problem, we need to define the decision variables, the objective function, and the constraints.
Decision Variables:
Let:
- X = number of gallons of Yodel beer to order
- Y = number of gallons of Shotz beer to order
- Z = number of gallons of Rainwater beer to order
Objective Function:
The objective is to maximize profit, which is calculated as the income minus the cost. The income is the selling price per gallon multiplied by the number of gallons for each brand, and the cost is the cost per gallon multiplied by the number of gallons for each brand.
The objective function is therefore:
Maximize: P = 3.00X + 2.50Y + 1.75Z - (1.50X + 0.90Y + 0.50Z)
Simplifying, we get: Maximize: P = 1.50X + 1.60Y + 1.25Z
Constraints:
1. Budget Constraint: The tavern has a budget of $2,000 for beer.
1.50X + 0.90Y + 0.50Z ≤ 2000
2. Demand Constraint: Based on past football games, the maximum customer demand is specified for each brand.
X ≤ 400
Y ≤ 500
Z ≤ 300
3. Stock Capacity Constraint: The tavern has a capacity to stock 1,000 gallons of beer.
X + Y + Z ≤ 1000
Additionally, since we cannot have negative quantities of beer, we also have the non-negativity constraints:
X ≥ 0
Y ≥ 0
Z ≥ 0
Therefore, the linear programming model can be summarized as follows:
Maximize: P = 1.50X + 1.60Y + 1.25Z
Subject to:
1.50X + 0.90Y + 0.50Z ≤ 2000 (Budget Constraint)
X ≤ 400 (Maximum demand for Yodel)
Y ≤ 500 (Maximum demand for Shotz)
Z ≤ 300 (Maximum demand for Rainwater)
X + Y + Z ≤ 1000 (Stock capacity constraint)
X ≥ 0, Y ≥ 0, Z ≥ 0 (Non-negativity constraints)
The objective function for this linear programming model is to maximize profit, which is represented as P. The profit is calculated by subtracting the cost of the beer from the income generated from selling the beer.
The decision variables in this problem are:
- X: the number of gallons of Yodel beer to order
- Y: the number of gallons of Shotz beer to order
- Z: the number of gallons of Rainwater beer to order
The cost of each brand per gallon is given as follows:
- Yodel: $1.50
- Shotz: $0.90
- Rainwater: $0.50
The selling price per gallon for each brand is given as follows:
- Yodel: $3.00
- Shotz: $2.50
- Rainwater: $1.75
The maximum customer demand for each brand is given as follows:
- Yodel: 400 gallons
- Shotz: 500 gallons
- Rainwater: 300 gallons
The tavern has a budget of $2,000 for beer. The tavern has a maximum capacity of 1,000 gallons.
The linear programming model can be formulated as follows:
Maximize P = 1.50X + 0.90Y + 0.50Z
subject to:
1.50X + 2.50Y + 1.75Z ≤ $2,000 (budget constraint)
X + Y + Z ≤ 1,000 (capacity constraint)
X ≤ 400 (customer demand constraint for Yodel)
Y ≤ 500 (customer demand constraint for Shotz)
Z ≤ 300 (customer demand constraint for Rainwater)
X, Y, Z ≥ 0 (non-negativity constraint)