Problem 2 (50 pts). An oil spil has fouled 200 miles of beach shoreline. The oil

Question

Problem 2 (50 pts). An oil spil has fouled 200 miles of beach shoreline. The oil company responsible is required to clean up the spill and is being fined $10,000 per day that the beach remains dirty. In order to clean up the sp the oil company will purchase a certain number of oil skimmer machines at a cost of See http://.papularnechanics.con/science/green-tech/a7162/spinning-disc-aysten-vins-oil-spil1-cleanup-challenge/ 8160 each, which are capable of cleaning 5 miles of shoreline per day. The oil company wants to determine the optimal umber of skimmer machines to purchase so as to minimize their overall cost Remark. In this problem, you are welcome to ignore integrality constraints; that is to say, you may assume that one is permitted to purchase "half a machine", or conversely, you can receive only a 85,000 fine if only half of a day is used. 1. Let x denote the number of skimmer machines that the company purchases and let d denote the number of days required for the cleanup to be completed. Write the total cost of cleanup (fines plus costs of skimmer machines) as a function of r and d. 2. Write an expression for d in terms of r, and use that to solve for the optimal number of skimmer machines a that the company should purchase. What is the total cost to the company in this case? 3. Determine the sensitivity of the optimal soluto the fine (S10,000) and the price of the machines ($160) using the same technique as in lecture. Which of the two prices has a greater impact on?

Explanation / Answer

So x = machines

d = days

Total cost = Fines+ expense of machine

= 10000*d + 160*x

2.

As total area cleaning

now 5xd=200

so xd= 40

d= 40/x

Total cost = Fines+ expense of machine

= 10000*d + 160*x

=10000*40/x + 160x

3.

So solving:

we get:

3.

Now if 1 unit of machine price is increased, the optimal value is

So increase of 49.92, hence sensitivity of 49.92

And

Now if 1 unit of fine is increased, the optimal value is

So increase of 00.79, hence sensitivity of 00.79

So higher sensitivity of machine as compared to fines.

Dec Var D Days 0.8 X Machines 50.0 Per machine cost 160 Per day Distance covered 5 Constraints (X*D*5) Total distance covered 200 = 200 Per day penalty 10000 D*10000 Fine 8000 X*160 Cost 8000 Obj Func Total expense 16000 Objective Func Minimize

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