Mountain Laurel Vineyards produces three kinds of wineMountain Blanc, Mountain Red, and Mountain Blush. The company has 17 tons of grapes
available to produce wine this season. Acask of Blanc requires 0.21 tons of grapes, a cask of Red requires 0.24 tons, and a cask of Blush
requires 0.18 tons. The vineyard has enough storage space in its aging room to store 80 casks of wine.
The vineyard has 2,500 hours of production capacity, and it requires 12 hours to produce a cask of Blanc, 14.5 hours to produce a cask of Red,
and 16 hours to produce a cask of Blush. From past sales the vineyard knows that demand for the Blush will be no more than half of the sales of
the other two wines combined. The profit for a cask of Blanc is $7,500, the profit for a cask of Red is $8,200, and the profit for a cask of Blush is
$10,500.
Formulate a linear programming model for this problem.
47. Solve the linear programming model formulated in Problem 46 for Mountain Laurel Vineyards by using the computer.
a. If the vineyard were to determine that the profit from Red was $7,600 instead of $8,200, how would that affect the optimal solution?
b. If the vineyard could secure one additional unit of any of the resources used in the production of wine, which one should it select?
c. If the vineyard could obtain 0.5 more tons of grapes, 500 more hours of production capacity, or enough storage capacity to store 4 more casks
of wine, which should it choose?
To formulate the linear programming model for this problem, we need to define the decision variables, objective function, and constraints.
Decision Variables:
Let x1, x2, and x3 represent the number of casks of Mountain Blanc, Mountain Red, and Mountain Blush produced, respectively.
Objective Function:
Maximize the total profit, which can be expressed as:
Profit = 7500x1 + 8200x2 + 10500x3
Constraints:
1. Grapes constraint: The total amount of grapes used cannot exceed the amount available.
0.21x1 + 0.24x2 + 0.18x3 ≤ 17
2. Aging room constraint: The total number of casks stored cannot exceed the storage capacity.
x1 + x2 + x3 ≤ 80
3. Production capacity constraint: The total production time cannot exceed the available production capacity.
12x1 + 14.5x2 + 16x3 ≤ 2500
4. Blush demand constraint: The demand for Blush cannot exceed half the sales of the other two wines combined.
x3 ≤ (x1 + x2) / 2
5. Non-negativity constraint:
x1, x2, x3 ≥ 0
Now, let's solve the linear programming model using a computer.
To solve using a computer program, we can use popular Linear Programming (LP) solvers like Excel Solver, MATLAB, R, or Python packages such as PuLP, CVXPY, or scipy.optimize.linprog.
Regarding the effects of changing profit for Red and the additional resources, we can analyze them one by one.
a. If the profit from Red changes from $8,200 to $7,600, the optimal solution may change because the objective function is affected. The solver will adjust the production quantities accordingly to maximize the profit within the new red wine profit margin.
b. To determine which resource should be acquired, we need to perform sensitivity analysis. By running the solver with an additional unit of each resource and comparing the resulting changes in the objective function value, we can determine which resource has the most significant impact on profit. The one with the highest increase in profit is the resource that should be selected.
c. Similar to part b, we can add each additional resource and compare the resulting changes in the objective function value to determine which resource provides the highest increase in profit. The resource that maximizes the profit increase should be chosen.
Note: The actual numerical values and optimal solution would require running the linear programming model using appropriate software.
To solve this linear programming problem, let's start by defining the decision variables and the objective function.
Decision Variables:
Let x1 be the number of casks of Mountain Blanc produced.
Let x2 be the number of casks of Mountain Red produced.
Let x3 be the number of casks of Mountain Blush produced.
Objective Function:
Maximize Profit = 7500x1 + 8200x2 + 10500x3
Constraints:
Grapes constraint: 0.21x1 + 0.24x2 + 0.18x3 <= 17
Storage constraint: x1 + x2 + x3 <= 80
Production capacity constraint: 12x1 + 14.5x2 + 16x3 <= 2500
Demand constraint: x3 <= 0.5(x1 + x2)
Now, let's solve this linear programming model using the computer.
a. If the profit from Red was $7,600 instead of $8,200, it would affect the objective function. The new objective function would be:
Maximize Profit = 7500x1 + 7600x2 + 10500x3
b. To decide which resource to select if the vineyard could secure one additional unit, we need to look at the shadow prices of the constraints in the optimal solution. The shadow prices represent the increase in profit for each additional unit of that resource.
c. To determine which additional resource to choose, we need to compare the shadow prices of the constraints related to grapes, production capacity, and storage capacity.
For the case of the vineyard producing more Blanc, Red, and Blush, the shadow prices represent the increase in profit for each additional ton of grapes, each additional hour of production capacity, and each additional cask of storage capacity, respectively.
Therefore, we need to compare the shadow prices for grapes, production capacity, and storage capacity to determine which resource to choose:
- 0.21 tons of grapes: Compare the shadow price for grapes with the shadow prices for production capacity and storage capacity. Choose to obtain 0.5 more tons of grapes if the shadow price for grapes is the highest.
- 500 hours of production capacity: Compare the shadow price for production capacity with the shadow prices for grapes and storage capacity. Choose to obtain 500 more hours of production capacity if the shadow price for production capacity is the highest.
- 4 more casks of storage capacity: Compare the shadow price for storage capacity with the shadow prices for grapes and production capacity. Choose to obtain 4 more casks of storage capacity if the shadow price for storage capacity is the highest.
By comparing the shadow prices, the vineyard should choose the resource with the highest shadow price to maximize profit and efficiency.