ECON 4004 A Fall 2022 Assignment 2
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ECON 4004 A Fall 2022
Assignment 2: Due November 22
PLEASE BE SURE TO READ THE GENERAL ASSIGNMENT GUIDELINES BEFORE YOU BEGIN THIS ASSIGNMENT.
1. Consider Example 11 of Section 3.7 involving the Semicond small electronics company.
a) Write down the dual problem. (Begin by converting the non-normal max primal problem to a normal max problem.)
b) Solve the dual problem by hand using the simplex algorithm. (Use the Big-M Method, if necessary, to obtain an initial tableau and then continue to use the Big-M Method as described on
p. 174 in Section 4- 12. Do NOT use the Two-Phase Simplex Method.) NOTE: Be sure to give a
complete written statement of the optimal solution to the dual problem.
c) Use the Dual Theorem, together with the CBV B −1 formula and the appropriate information from your answer to part b), to deduce the optimal solution to the primal problem. NOTE: Be sure to give a complete written statement of the optimal solution to the primal problem and to show all of your working.
2. Affordable Gifts Inc. produces three types of paper decorations: Cats, Dogs, and Rabbits. The sales prices of these decorations are $6, $8, and $13, respectively. To produce a Cat takes 3 hours of labour and 2 sheets of paper, to produce a Dog takes 4 hours of labour and 2 sheets of labour, and to produce a Rabbit takes 6 hours of labour and 5 sheets of paper. At present 60 sheets of paper are available. Up to 90 hours of labour can be purchased at an hourly wage-rate of $1.
a) Formulate an LP that could be used to maximize profits. NOTE: Be careful to ensure that all your constraints are in the correct format, namely, a linear function of the decision variables on the left-hand side of a ≤ , =, or ≥ sign and a number on the right-hand side.
b) Solve this LP using LINDO. Copy and paste the LINDO output into your assignment, including the problem statement, the regular output, and the sensitivity analysis. Also, be sure to highlight the optimal solution.
c) Use your LINDO output from part b) to solve the following questions:
i) What is the most that Affordable Gifts should pay for another sheet of paper? Explain.
ii) What is the most that Affordable Gifts should pay for another hour of labour? Explain.
iii) What would the Cat have to sell for in order to make it desirable for Affordable Gifts to produce it? Explain.
iv) If 100 hours of labour could be purchased, rather than 90 hours, what would be the profit of Affordable Gifts? Explain.
v) What would the new optimal solution be if the Rabbit sold for $15? Explain.
3. The CEO of a local TV station wants to determine the optimal way of allocating the 20 minutes of broadcast time available for the early evening news. More specifically, he would like to allocate the available time across four distinct areas, namely, local news, entertainment, sports, and weather. Federal broadcast regulations stipulate that at least 15% of the time available must be devoted to local news, the time devoted to local news and/or entertainment must be at least 50% of the time available, the time devoted to weather coverage must be less than or equal to that devoted to sports coverage, the time devoted to sports coverage must be no more than the total time spent on local news and/or entertainment, and at least 20% of the time must be devoted to weather coverage. Per minute production costs are running at $300 for local news, $200 for entertainment, $100 for weather, and $100 for sports.
a) Formulate an LP that could be used to maximize profits. NOTE: Be careful to ensure that all your constraints are in the correct format, namely, a linear function of the decision variables on the left-hand side of a ≤ , =, or ≥ sign and a number on the right-hand side.
b) Solve this LP using LINDO. Copy and paste the LINDO output into your assignment, including the problem statement, the regular output, and the sensitivity analysis. Also, be sure to highlight the optimal solution.
c) Carefully interpret the dual price for the available time constraint.
d) Carefully interpret the dual price for the constraint requiring that at least 15% of the time available be devoted to local news.
e) Carefully interpret the dual price for the constraint requiring that at least 50% of the time available be devoted to local news and/or entertainment.
f) Carefully interpret the dual price for the constraint requiring that the time devoted to weather coverage must be less than or equal to the time devoted to sports coverage.
2022-11-16