Problem 1: Model the problem using Decision Vars. (DV), Obj. Function (OF), and Constraints (C). A company faces the following demands during the next three periods: period 1, 20 units; period 2, 10 units; period 3, 15 units. The unit production cost during each period is as follows: period 1$13; period 2$14; period 3$15. A holding cost of $2 per unit is assessed against each periods ending inventory. At the beginning of period 1, the company has 5 units on hand. In reality, not all goods produced during a month can be used to meet the current months demand. To model this fact, we assume that only one half of the goods produced during a period can be used to meet the current periods demands. Formulate an LP to minimize the cost of meeting the demand for the next three periods. (Hint: Constraints such as 1 = 1 + 5 20 are certainly needed. Unlike our example, however, the constraint 1 0 will not ensure that period 1s demand is met. For example, if 1 = 20, then 1 0 will hold, but because only (20) = 10 units of period 1 production can be used to meet period 1s demand, 1 = 20 would not be feasible. Try to think of a type of constraint that will ensure that what is available to meet each periods demand is at least as large as that periods demand.)

Problem 2: Model the following problem (DV, OF, C) and answer the logistics question. Granjas Trigo-Maiz has two farms that grow wheat and corn. Because of differing soil conditions, there are differences in the yields and costs of growing crops on the two farms. The yields and costs are shown in the table below. Each farm has 100 acres available for cultivation; 11,000 bushels of wheat and 7,000 bushels of corn must be grown. Determine a planting plan that will minimize the cost of meeting these demands. How could an extension of this model be used to allocate crop production efficiently throughout a nation? China farms Valle Hermoso farms Corn yield/acre (bushels) 500 650 Cost/acre of corn ($) 100 120 Wheat yield/acre (bushels) 400 350 Cost/acre of wheat ($) 90 80

Problem 3: Model the following problem, again clearly state DV, OF and C. The city of The Path contains three school districts. The number of minority and nonminority students in each district is given in the table below. Of all students, 25% (200/800) are minority students. District Minority Students Nonminority students Coronado (distance km) Americas (distance km) 1 Canutillo 50 200 1 2 2 Socorro 50 250 2 1 3 Ysleta 100 150 1 1 The local court has decided that both towns two high schools, Coronado and Americas, must have approximately the same percentage of minority students (within 5%) as the entire town. The distances (in kms) between the school districts and the high schools are given in the last two columns. Each high school must have an enrollment of 300500 students. Use linear programming to determine an assignment of students to schools that minimizes the total distance students must travel to school.

Problem 1 & 3 asks you to 1. Model the problem Problem 2 asks you to 2. Model the problem 3. Answer the following question: How could an extension of this model be used to allocate crop production efficiently throughout a nation? FOR ALL problems, when you model a problem, make sure you have: A. Decision Variables a. Properly defined B. Objective Function a. Units match C. Constraints a. Properly defined b. Domain