Question: For the linear programming problem:
Maximize 10X ₁ + 6X ₂ s.t.
①  2X ₁ + 3X ₂ ≤ 36;
②  5 X ₁ + 3 X ₂ ≤ 45;
③ 15X ₁ + 4 X ₂ ≤ 120;
X ₁ ≥ 0; X ₂ ≥ 0.

1. Draw a graph of this linear programming problem, with each constraint line labeled and the feasibility region shaded.
2. Give a reasonably plausible interpretation of this problem. Include as part of that interpretation an overall context and a specific "real-life" meaning for 𝑥 and 𝑦, for each of the constraints, and for the objective function.
3. Graphically solve this problem, finding the optimal set of (X ₁ , X ₂ ) and the value of the objective function. Which constrain(s) are binding?
4. What are the last two constraints in the problem called?
5. For the simplex algorithm, what is the value (X ₁ , X ₂ ) to be tried first as a potential optimal solution?
6. If the point in (e) is not optimal, list all possible candidate points (X ₁ , X ₂ ) that could be considered as a trial solution in the second iteration of the simplex algorithm. (Note: The actual second trial value for this problem is not to be found by working through the simplex algorithm, but rather all possible second trial points based on knowledge of general properties as to how the simplex algorithm works.
7. Use the simplex algorithm to solve this problem, showing all key steps of the algorithm, such as each tableau, knowing when the final iteration has been completed versus at least one more iteration needs to be done and, if the latter case how to determine what value of (X ₁ , X ₂ ) is to be tried next.
8. Do both techniques yield the same optimal solution set? Explain.
9. How would the optimal solution set change if the objective function for maximization changed to 10X ₁ + 7X ₂ ?

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