Formulate this problem mathematically. In lines entered below this part, state your complete mathematical representation of this problem. Your complete model must (i) define all the decision variables, (ii) state the objective function in terms of your decision variables, and (iii) state all of the constraints in terms of your decision variables.
The Mason Wine Company produces two kinds of wine – Mason Blanc and Mason Merlot. The wines are produced in 1,000-gallon batches. The profit for a batch of Blanc is $12,000 and the profit for a batch of Merlot is $9,000.
The wines are produced from 64 tons of grapes that the company has acquired. A 1,000-gallon batch of Blanc requires 4 tons of grapes and a batch of Merlot requires 8 tons.
However, the production is limited by the availability of only 50 cubic yards (yd3) of storage space for aging and 125 hours of processing time. Each batch of each type of wine requires 5 yd3 of storage space. The processing time for a batch of Blanc is 15 hours and the processing time for a batch of Merlot is 7 hours.
The wine company will not produce more or less than the range of amounts demanded for each type. Demand for each type of wine is for at least three batches but is limited to not more than eight batches. Even so, the demand for Blanc is the same as or is higher than the demand for Merlot.
Company executives do not want to depend on just one type of wine so they have mandated minimum production levels of both types of wine. Specifically, at least one-fourth of the total wine production must be Merlot. Likewise, at least a quarter of the total wine production must be Blanc. Moreover, the production (in batches) of Blanc must be at least twice the production of Merlot.
Also, the break-even point on profit is $54,000. Therefore, company management requires that it must make at least $54,000 in profit to do better than just break even (that is, profit must be $54,000 or more).
The company wants to set the production levels in terms of the number of 1,000-gallon batches of both the Blanc and Merlot wines to produce so as to earn the most profit possible.
(NOTE: Partial [i.e., fractional] batches can be produced; they refer to pending production. If fractional, round the optimal solution values to three (3) decimal places.)
-Questions:
1. Formulate this problem mathematically. In lines entered below this part, state your complete mathematical representation of this problem.
Your complete model must (i) define all the decision variables, (ii) state the objective function in terms of your decision variables, and (iii) state all of the constraints in terms of your decision variables.
2. In lines entered below this part, state the specific type of problem this is. Include an explanation of why.
3. a. Graph this problem consisting of the following specific steps:
Graph (and label) all of the constraints.
On your graph, identify and label the feasible region.
b. Identify the optimal solution point using the contour sweep method consisting of the following specific steps:
On your graph, plot contours of the objective function and label each with its value.
On your graph, identify and label on your graph the optimizing direction.
On your graph, identify and label on your graph the point containing the optimal solution.
In a space created below this part, present HERE a copy of your labeled graph.
4. a. In lines entered below this part, state why the optimal solution is located at the point identified in your graph. That is, state why the contour sweep method led you to conclude that this specific point contains the optimal solution.
b. In lines entered below this part, state the location of the optimal solution point in terms of where it is in the feasible region.
c. In lines entered below this part, state briefly the methodology you will use to calculate the optimal values of the decision variables.
d. Based on part 3, calculate and explicitly state in lines entered below this part the numerical values of the optimal solution in terms of the optimal values of the decision variables and the optimal value of the objective function.
NOTE: You MUST show your work HERE to receive credit for part 4c.
5. In lines entered below this part, explicitly state in plain language the optimal numbers of batches of Merlot and Blanc wines to make and the maximum amount of the profit.
6. Determine the optimal solution with the corner point evaluation method.
a. In lines entered below this part, calculate and explicitly state the values of the decision variables and of the objective function at all the corner points of the feasible region identified in graph in part 3.
b. In lines entered below this part, calculate and explicitly state the optimal solution and optimal value calculated by the method based on your results in part 6a. You must show those calculations.
c. In lines entered below this part, compare the results in parts 4 and 6b. Are they the same or not? Either way, why?
7. a. Calculate and explicitly state in lines entered below the amounts of slack/surplus for each and every constraint of your model.
b. In lines entered below this part, state in plain language the specific meaning of the numerical value slack/surplus for each and every constraint of your model in the context of the business situation. Be specific and complete.
8. In lines entered below, identify all the active, all the inactive, and all the redundant (if any) constraints. In other words, classify each and every constraint in your model as to whether it is active, inactive, or redundant.
9. Solve this problem with a computer using LINGO.
In a space created below this part, present HERE copies of both the computer input and output screens.
Submit all of your computer files with your completed exam.
10. In lines entered below, explicitly state the optimal solution (in terms of the optimal values of the decision variables and of the objective function) that was calculated and stated on the computer output presented in part 9.
11. In lines entered below, compare your results in part 10 value with your previous results in parts 4 and 6b. In particular, state whether the results are the same or not. Include an explanation of why or why not.
12. In lines entered below, calculate and explicitly state the ranges of optimality on the profit per batch for each type of wine (that is, the ranges of optimality for each of the coefficients of the objective function stated as ranges [not as allowable increases/decreases]).
13. In lines entered below, calculate and explicitly state the ranges of feasibility of all the constraints (that is, the ranges of feasibility of the right-hand sides of all the constraints stated as ranges [not as allowable increases/decreases]).
14. In lines entered below, state in plain language the specific meaning of the numerical values of each of the non-zero dual prices in the context of this business situation. In other words, state what each non-zero numerical value of a dual price means in the context of this business situation stated on page 2. Be explicit and specific.
15. Suppose that it is now determined that the break-even point has risen by $50,000 more than was originally stated.
a. In lines entered below, explain how the feasible region changes, if at all.
b. In lines entered below, explain how the optimal solution and the optimal value are changed, if at all.
Requirements: All answers in Excel and Q9 in LINGO
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