Your club attaches its logo to various clothing items and distributes them to its members. Since 10 patches were left over after doing this, you decide also to attach these to clothing items and sell them to the general public for a profit. The clothing items you can get for a very good price: $2.00 for each sweatshirt, $1.00 for each T-shirt, and $2.00 for each ball cap. But unfortunately, your club is down to almost no funds, so you have only $12.00 left in the budget for purchasing these additional unmarked clothing items. The profit come to $3.00 for each sweatshirt, $2.5 for each T-shirt, and $4.00 for each ball cap. In order to improve the funding situation for your club as much as possible, your goal is to maximize the profit made from the sale to the public of these additional clothing items.
Part a) Formulate this as a linear programming problem, stating whatever assumption you make in order to consider it an LP.
Part b) Solve this problem using software.
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Part c) State the dual problem to this problem.
Part d) Solve the dual model using software.
Part e) It turns out that increasing the number of patches left over from 10 to 11 will not change the optimal basis in the end. Knowing this what would be overall profit from the sale of these items be if we were to find we had 11 patches left over, rather than 10 as originally thought? Explain your reasoning based on duality theory.