Mountain Laurel Vineyards produces three kinds of wine Mountain Blanc, Mountain Red, and Mountain Blush. The company has 17 tons of grapes available to produce wine this season . A cask 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.
49. 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?
d. All three wines are produced in the optimal solution. How little would the profit for Blanc have to be for it to no longer be produced?
To formulate the linear programming model for this problem, we need to define the decision variables, objective function, and constraints.
Decision Variables:
Let B represent the number of casks of Blanc wine
Let R represent the number of casks of Red wine
Let U represent the number of casks of Blush wine
Objective Function:
Maximize Profit = 7500B + 8200R + 10500U
Constraints:
1. Grape Constraint: 0.21B + 0.24R + 0.18U ≤ 17 (tons of grapes available)
2. Aging Room Constraint: B + R + U ≤ 80 (casks of wine)
3. Production Capacity Constraint: 12B + 14.5R + 16U ≤ 2500 (hours of production capacity)
4. Demand Constraint: U ≤ (B + R) / 2 (demand for Blush)
Now, we can proceed to solve the linear programming model using a computer.
a. If the profit from Red was reduced to $7,600, the optimal solution may change. To determine the exact impact, the linear programming model needs to be solved again with the updated profit value for Red.
b. To determine which resource to choose for an additional unit, we need to consider the shadow prices of the constraints from the optimal solution. The constraint with the highest shadow price indicates the resource that is most valuable to secure an additional unit.
c. To determine which option to choose for the additional resource, we need to evaluate the impact of each option on the objective function. We can calculate the new profit values based on the additional resource and compare the results.
d. If all three wines are produced in the optimal solution, we need to find the point at which the profit for Blanc becomes zero. This can be done by adjusting the profit coefficient for Blanc in the objective function and solving the linear programming model iteratively until its profit value reaches zero.