AGEC 7100 Old final Exam Name ____________________________ 1. Maximize Profit = 5X1 + 3X2 + 4 X3 + 2X4 subject to: 2X1 + X2 + 3X3 + X4 ≤ 10 3X1 + X2 + X3 ≤ 15 X4 ≤ 2 a) Set up the simplex tableau for this problem, including all required slack variables. (The space below may be larger than the space required.) b) What is the greatest number of decision variables (X variables) that can be basic? How do you know? 2. Look at the tableau below for a minimization problem. Cost X1 X2 X3 S1 S2 M1 M2 RHSobj 1 120.75 0 172.25 -100 24.75 0 -124.75 1628.75r1 0 1.25 0 1.75 -1 0.25 1 -0.25 16.25r2 0 0.75 1 0.25 0 -0.25 0 0.25 3.75 a) Is this solution optimal? How do you know? b) Read the current solution from the tableau (decision variables and cost) c) Circle the pivot element.3. Look at the tableau below, for a maximization problem. Prof x1 x2 x3 s1 s2 s3 rhsobj 1 1.5 0 0.25 0 2.25 0 40.5r1 0000.51-0.501r2 0 0.5 1 0.25 0 0.25 0 4.5r3 0 -0.5 0 -0.25 0 -0.25 1 2.51 a) Is this solution optimal? How do you know? b) Read the current solution (decision variables and costs) from the table. c) Now give the shadow prices (duals) on the two constraints. d) Now list the reduced costs on the activities. 4. Set up the dual to the problem in 1, above.5. Consider the problem: Maximize Profit = 20 X1 + 9 X2 + X3 Subject to: 3X1 + X2 GE 20 X1 - 2X2 + X3 = 40 2X1 + X2 LE 90 Where: X1 GE 0, X2 LE 0, and X3 unrestricted in sign. a) Directly take the dual of this problem. b) Convert this problem to a standard "max" by substituting variables and "fixing" the constraints.6. A producer may produce cattle (net profit $1500 per 50 cows), cotton (net profit ($50/acre) or soybeans ($40/acre). She has 600 acres of cropland, 200 acres of pasture, 1000 hours of spring labor, and 4000 hours of fall labor. Each acre of cotton requires 2 hours spring labor, and 3 hours harvest labor. Each acre of soybeans requires 2 hours spring labor and 3 hours fall labor. A 50-cow cattle unit requires 100 acres of pasture, 2 acres of cropland, 200 hours of spring labor and 80 hours of fall labor. a) In the space below, set up the LP tableau (for computer solution) for the problem. Be sure to label the constraints in the space provided. Cattle Cotton Soybeans Type RHS b) Look at the resource requirements and availability. Which resource would not be limiting in this problem? c) Could soybeans enter into the optimal solution? Why or why not? 7. Here is an LP tableau and its solution. Cotton Corn Soybeans Cattle Shadow1 acre 1 acre 1 acre 50 cows Type RHS Used priceobj 80 60 50 2300 max 59320cropland1110le80080035pasture 0 0 0 100 le 120 120 11spring lab 2 2.5 1 80 le 2000 1448 0fall lab 3 2 1 80 le 2000 2000 15answers 552 0 248 1.2red cost 0 -5 0 0 Interpret this output.Use the paper provided to answer questions 8 and 9. 8. A company produces cut-up chickens from whole chickens. Tasks must be performed in the following order: gutting, washing, cutting. Gutting takes 0.05 hour of labor per chicken, washing takes 0.01 hour of labor per chicken, and cutting takes 0.10 hour of labor per chicken. There are 200 hours of labor available in each hour, and four hours in each shift. Any chickens not completely processed (e.g. cut up) at the end of the four hour shift must be dumped. The cut up chicken packs sell for $2.50 each. Set up a sequencing problem for a four hour shift. (Assume labor can move freely from task to task whenever needed.) 9. Set up an equilibrium unknown-life tableau for the following problem involving corn and a vegetable enterprise: Cornyear 1 Year 2 Year 3 Year 4 any yearReturns -100 500 300 200 300Labor requirements 2 2 0.5 0.5 1Land requirements 11111Vegetables There are 400 acres of land available and 600 hours of labor. 10. Derive the Kuhn-Tucker Conditions for the following problem: Maximize: C= 9X1 + 12X2 -X12 - 2X22 s.t. X1 LE 20 3X1 + 5X2 LE