Question 1. A farmer has 500 acres of land and wishes to determine the acreage allocated to the fallowing three crops: wheat, corn and soybeans. The man-days required, preparation cost. and profit per acre of the three crops are summarized below. Suppose that the maximum number of man-days available are 5,000, and that the farmer has $70,000 for preparation.
| Crop |
Man-Days |
Preparation Costs ($) |
Profits ($/acre) |
| Wheat |
6 |
100 |
60 |
| Com |
10 |
150 |
90 |
| Soy Beans |
8 |
120 |
80 |
Formulate a linear program thal maximizes the farmer's profit.
Question 2. A power utibty has a number of power stations of three different types that they use to provide power to a s,ific region. 12 of tyce 10 of type 2, and 5 of tyce 3. Each plant has to run between a minimum and maximum level. T. hourly cost of running a plant is calculatce as a bace cost (for running at rninimum power). an hourly cost for cach additional megawatt over the minimum. plus an additional cost for starting a generator. All of this information is contained in the following table (where costs are in dollars).
| Power Plant Type |
Minimum operating Level |
Maximum operating Level |
Cost per Hour at Minimum |
Cost per Hour per magwatt above Minimum |
Startup Cost |
| 1 |
1,200 MW |
2,000 MW |
1000 |
2 |
2,000 |
| 2 |
1,050 MW |
1,500 MW |
2600 |
1.3 |
1,000 |
| 3 |
2,500 MW |
4,000 MW |
3000 |
3 |
500 |
The demand for power throughout dce day in this region is given below, but the utility must be able to provide up to 15% more power at any point from i. running plants without exceeding their maximum limits,
12am to 6am 15,000 MW
6am to 9am 30,000 MW
9am to 3, 25,000 MW
3pm to 6, 40,000 MW
6pm to I.m 27,000 MW
(a) Write a linear model that detennince the minimum cost generation plan.
(b) What is the marginal cost of power production for each of the 5 periods of the day. That is. what is the minimum amount that should be charged for power in each time period,
(c) What would be the savings generated by lowering the 15% peak power reserve? That is, what is the cost of this safety margin? Copy and modify your model to find this answer.