Faaliyetler İş Araştırma ve İktisat Homework 1 fatihbook.com fatihbook is not responsible for any mistakes 1 Man 203 Quantitative Methods, Nizamettin Bayyurt HW1, Due date: 24 Nov., 2007 1. Put the following LP problem in standard form: Minimize Z=2x1+4x2, subject to x1+x2 ? 4 x1+x2 ? 1 x1,x2 ? 0 2. Consider the following linear programming problem. Maximize Z=40x1+60x2, subject to 2x1+x2 ? 7 x1+x2 ? 4 x1+3x2 ? 9 x1,x2 ? 0 (a) Plot the feasible region. Label all lines and their intersections. (b) Solve the LP problem using the “corner point” method. Clearly state the optimal value and optimal solution. 3. Show that the following LP problem is infeasible. Maximize Z=3x1 -x2, subject to x1+x2 ? 2 -2x1-2x2 ? -10 x1,x2 ? 0 4. Show that the following LP problem is unbounded. Maximize Z=x1-x2, subject to -2x1+x2 ? -1 -x1-2x2 ? -2 x1,x2 ? 0 5. Solve the problem by using the Simplex method Maximize Z=2x1+3x2+3x3, subject to 3x1+x2 ? 60 -x1+x2+4x3 ? 10 2x1-2x2+5x3 ?15 and x1,x2,x3 ? 0 6. Tucker Inc. Needs to produce 1000 Tucker automobiles. The company has four production plants. Due to differing workforces, technological advances, and so on, the plants differ in the cost of producing each car. They also use a different amount of labor and raw material at each. This is summarized in the following table: fatihbook.com fatihbook is not responsible for any mistakes 2 Plant Cost (‘000) Labor Material 1 15 2 3 2 10 3 4 3 9 4 5 4 7 5 6 The labor contract signed requires at least 400 cars to be produced at plant 3; there are 3300 hours of labor and 4000 units of material that can be allocated to the four plants. Write the model of this problem. For the questions 7-12 Use the table given below as the final table of a maximization problem. X1 X2 X3 S1 S2 Cj 2 3 1 0 0 X1 2 1 0 1/4 -3 7 105 X2 3 0 1 1/6 4 -2 160 Zj 2 3 1 6 8 590 Cj-Zj 0 0 0 -6 -8 7. What are the current production quantities? What is the current profit of production? 8. Are the alternative solutions? Why? 9. How the changes of the profit of X3, decreased to 0.8 from 1, effect the solution mix and total profit? 10. What are the shadow prices? 11. If the RHS value of the first constraint increased by 45 units how is the optimal value affected from this? 12. If the RHS value of the second constraint decreased by 40 units how is the optimal value affected from this? 13. Write the dual form of the problem MaxZ= 3x1+2x2 X2 ? 6 3x1+2x2 ? 18