A utility company supplies drinking water to a community, which demands at least 10,000 m3 of water daily, from two pumping stations (A and C). Station A has a maximum supply capacity of 18,000 m3 , while station C has a maximum capacity of 20,000 m3 per day. Due to operational constraints, station C must supply at least 6,000 m3 of water. In addition, since both stations handle different purity levels, the water authority stated that the amount of water supplied by station A must be at least one-third of the amount supplied by station C. The operational costs of supplying one cubic meter (m3 ) of water from the stations are 2000 ($/m3 ) from A and 1000 ($/m3 ) from C. Please formulate a linear programming model that allows finding the optimal way to meet the communitys water demand at the lowest possible cost, while complying with the systems requirements. The following questions need to be answered (elaborate on your answers):

1. (10 points) Formulate the optimization model for this problem: define the decision variables, objective function, and constraints;

2. (5 points) Solve the problem graphically: draw the feasible region, evaluate its corner points, and point out the optimal solution;

3. (5 points) Classify the constraints into active and non-active.

4. (10 points) The company is considering expanding the supply capacity of one of the two stations and wants you to make a recommendation in this regard; Write: which station to expand and by what amount?

5. (10 points) How much would the total costs increase if the capacity of station A were reduced by half, that is, 9000 m3 , and explain why?

6. (10 points) Suppose that the cost of supplying water from station C exceeds that of A, reaching $2500 ($/m3 ); should the optimal amounts supplied from both stations be changed? Write: yes or no, why, and the new optimal amounts to be supplied from both stations.

7. (10 points) What would be the dual or shadow price of the constraint representing the maximum supply capacity of station A, and why?

8. (10 points) What would be the allowable increase for the dual price of such a constraint mentioned in question 7, and why?

9. (15 points) What would be the solution to the problem if the maximum supply capacity of station C were 6000 m3 and the water authority required that the amount supplied by station A were at most one-third of the amount supplied by station C, and why? (note: assume that all the other elements of the problem remain unchanged).

10. (15 points) Describe the special case we would be facing if the operational costs of both pumping stations, A and C, were the same, and why?