Question

a) Formulate a cost function along with constraints, if any, for the following optimization problems. You don't need to solve any of these problems i) Two electric generators are interconnected to provide total power to meet the load. Suppose each generator's cost (C) is a function of its power output P (in terms of units), and costs per unit are given by: C2 = 1 + 0.6P2 + P22 (for Generator 2). -1-P -Pi2 (for Generator 1), If the total power needed is at least 60 units, formulate a minimum-cost design problem to determine the power outputs Pi and P2 from the two units ii) Points A and B are opposite each other on the shores of a straight river that is 1.5 miles wide. Point C is on the same shore as B but 3 miles down the shoreline of the river from B. A telephone company wishes to a lay a cable from A to C. Of course, a section of the cable has to be routed under the water, with the remaining section laid on the land. If the cost of laying a cable on land $10,000/mile, whereas doing same under water costs $30,000/mile, find an optimal cable laying strategy that will minimize the overall cost. (Hint: find a suitable point, x, between B and ) Consider a small farm that produces 5 to 15 units of a product daily, and the profit per unit is iven by 100 - (x-5)2, where x denotes the number of units produced. Using classical optimization chnique, find the optimal value of x that maximizes the total daily profit. Also, please verify that our solution corresponds to a maximum.

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