MATHEMATICS II: COLLECTION OF EXERCISES AND PROBLEMS

MATHEMATICS II: COLLECTION OF EXERCISES AND PROBLEMS GRADO EN A.D.E. GRADO EN ECONOMÍA GRADO EN F.Y.C. ACADEMIC YEAR 2011-12

INDEX UNIT 1.- AN INTRODUCCTION TO OPTIMIZATION 2 UNIT 2.- NONLINEAR PROGRAMMING 5 UNIT 3.- AN INTRODUCTION TO LINEAR PROGRAMMING 15 UNIT 4.- THE SIMPLEX METHOD 19 UNIT 5.- DUALITY IN LINEAR PROGRAMMING 24 UNIT 6.- SENSITIVITY ANALYSIS AND POST-OPTIMIZATION 30 UNIT 7.- INTEGER LINEAR PROGRAMMING 37 1

UNIT 1.- AN INTRODUCTION TO OPTIMIZATION 1. Identify the set of variables, the objective function, the set of constraints, the sign conditions and the solution set of the following Mathematical Programming problem. a) Min. x + y s. t. 2x + y 3 x, y 0 b) Max. x 2 + y 2 + z 2 s. t. x + y 6 x + 3y 8 x + 2y + z 20 x, y 0 2. Draw two level curves for each of the following functions. a) f(x, y) = 3x 2y b) g(x, y) = x 2 + y 2 c) h(x, y) = x + 5y 3. Draw the following sets and say if they are closed and/or bounded sets. a) S = {(x, y) R 2 / x + y 6, 2x + y 3, x 0, y 0} b) S = {(x, y) R 2 / x + y 6, x + 2y 3, y 0} c) S = {(x, y) R 2 / x 2 + y 2 9} 4. Solve the following problems graphically I) Max. x + y s. t. 2x + y 3 x + 2y 3 x + y 6 x 0, y 0 II) Max. 2x + 3y s. t. x + 2y 10 x 20 x 0, y 0 III) Max. x y s. t. 3x + 2y 5 x 0, y 0 IV) Max. x 2 + y 2 s. t. x + y = 1 V) Min. x 2 + y 2 s. t. x + y = 1 VI) Max. x + y s. t. x + 4y 5 x 0, y 0 2

5. Transform the problems of exercise 4 into their canonical form (maximization objective with less or equal tan inequalities, or minimization objective with greater or equal than inequalities) and into standard form (equality constraints and non-negative variables). 6. Give an example of each of the following cases: a) A problem having optimal solution whose solution set is bounded. b) An unbounded problem whose solution set is bounded. c) A problem with optimal solution whose solution set is unbounded. d) An unbounded problem whose solution set is unbounded, too. 7. Prove the existence of global optimum for the following problems theoretically: I) Max. x + y s. t. 3x + y 4 x + y 5 x 0, y 0 II) Max. x 2 + 3y 2 s. t. (x 1) 2 + (y 3) 2 4 x 0, y 0 III) Min. x y s. t. 3x + 2y 5 x 0, y 0 8. Find for the following problems a feasible interior solution and a feasible boundary solution (if there are any). I) Min. x y s. t. x + y 5 x 0, y 0 II) Max. x + y s. t. x = 4 x 0, y 0 III) Max. x + 3y s. t. (x 2) 2 + (y 1) 2 9 x 0, y 0 IV) Min. x + 3y s. t. y = x 2 x 0, y 0 9. Apply the Local-Global theorem for each of the following cases: a) We know that (11,10) is a local minimum of the following problem: Min. x 2 + y 2 + 2x + 4y + 10 s. t. x + y = 21 b) We know that (3,3) is a local maximum of the following problem: Max. x + y + 14 s. t. x + y 6 x 0 c) We know that (7,0) is a local minimum of the following problem: 3

Min. x 2 y 2 s. t. x + 5y 7 x, y 0 10. Let P be a Mathematical Programming problem, and let P be another problem resulting from adding one more constraint to P. Let us suppose that its objective is to maximize. a) Which solution set will be larger, that of P or of P? b) Which optimal solution will be higher, that of P or of P? c) If x is a feasible solution of P, will it be feasible for P too? And the other way around? d) If x* is the optimal solution of P, will it be optimal for P too? 11. Let us suppose that we have solved a Mathematical Programming problem with constraints and we have found a global optimum. If we remove the constraints, will the problem still have optimum? And if it has, will it be better or worse than that of the problem with constraints? Can it be the same? 12. A maximization problem has optimal solution and we have found a solution for which the objective function has value 25. Can we state that the optimal value of the objective function is greater than 25? And greater or equal? 13.- Calculate, when possible: a) {1,3,6} {2,5,6} b) {1,3,6} {2,5,6} c) {(0,1),(3,4)} {(0,4)} d) {(0,2,-6),(8,-1,4,8)} R 4 e) {(x,y) R 2 / x 2 +y 2 9} {(x,y) R 2 / x 2 +y 2 14} f) {(x,y) R 2 / x 2 +y 2 9} {(x,y,z) R 3 / x 2 +y 2 14} g) {(x,y) R 2 / x+2y 4} {(x,y) R 2 / 3x+y 3} h) {x R / x 14} {y R / y 3} i) ([3,+ [ [2,4]) {x R / x 14} 14.- Add the appropriate symbols in order to make the following statements true. You can use:,,,,,, <, >,. 4

a) 1 {6,1,4} b) [2,8] [-3,8] [2,12] c) {x R / 3 x 18} = [3,18] d) {(x,y) R 2 / x + y 13, 2 x 3 y 13 } = = {(x,y) R 2 / x + y 13 } {(x,y) R 2 / 2 x 3 y 13 } e) [-2,9] {x R / 3 x 9} [-2,7] f) { x R / x 13} [2, [ g) [2,8] [9,14] = 15.- Study graphically if the following sets are convex: a) {(x,y) R 2 / x+2y 4} b) {(0,1),(3,4)} c) {(3,-1)} d) {(x,y) R 2 / -5x+y=4} e) {(x,y) R 2 / x+y 4, x 0, y 0} f) {(x,y) R 2 / x 2 +y 2 4, y 0} g) {(x,y) R 2 / x y 2 } 16.- Study if the following functions are convex and/or concave: a) f(x,y)=x 2 +y b) f(x,y,z)=x-3y+2z c) f(x,y)=-4x 2 -y 2 d) f(x,y)=3x 2 +2y 2 +4xy-6 e) f(x,y,z) = 3x 2 +2y 2 +3z 2-2xy+4xz f) f(x,y,z) = -4x-2y 2 +xy+yz g) f(x,y,z) = -2x-y 2-3xy-5yz h) f(x,y) = ln(x+y) i) f(x,y) = e x+y 17.- Using the properties of convex sets, study if the following sets are convex or not: a) {(x,y) R 2 / x+2y 4, 3x-2y 7} 5

b) {(x,y) R 2 / 3x 2 +2y 2 +4xy-6 30} c) {(x,y) R 2 / -4x 2 -y 2 7} d) {(x,y) R 2 / ln(x+y) 25, -5x+3y 10, y 0} e) {(x,y) R 2 / 3x 2 +2y 2 +4xy-6 30} {(x,y) R 2 / ln(x+y) 25} 6

UNIT 2.- NON-LINEAR PROGRAMMING 1. Write the Kuhn-Tucker conditions of the following problems: I) Min. x 2 + y 2 s. t. : x + y 1 II) Max. x s. t.: x 2 + y 2 1, x 0, y 0 III) Max. x 1 2 + y 1 2 s. t.: x 2 + y 2 1 IV) Max. ln(x + y) s. t. : 6 x 2 y 2 0, x 2 y 0, y 0 V) Min. x + 2y + 4z s. t. : xy 2 z 2 = 120, x 0 2. Obtain all the Kuhn-Tucker points of the following problems: I) Min. x 2 + y 2 s. t. : x + y 1 II) Min. x 2 + (y 1) 2 s. t. : x + y 8, 2x + y 4 III) Min. xy s. t. : x y 0, x 2 + y 2 2 3. Analyze the constraint qualification in the following solution sets: I) S = {(x, y) R 2 /x + y 1, x y 0, x 0} II) S = {(x, y) R 2 /x 2 + y 2 1, x 0} III) S = {(x, y) R 2 /x 2 + y 2 2, x y 0} IV) S = {(x, y) R 2 / (x 1) 3 y 0, y 0} 4. For each one of the following problems, say what each function f, g 1, and g 2 must be (concave, convex, or lineal) in order to fulfill the hypothesis of the Kuhn-Tucker sufficiency theorem: I) Max. f(x, y) s. t.: g 1 (x, y) b 1, g 2 (x, y) = b 2, y 0 II) Min. f(x, y) s. t.: g 1 (x, y) b 1, g 2 (x, y) = b 2 III) Max. f(x, y) s. t.: g 1 (x, y) = b 1, x 0, y 0 IV) Min. f(x, y) s. t.: g 1 (x, y) b 1, g 2 (x, y) b 2, x 0 5. Analyze the hypothesis of the Kuhn-Tucker sufficency theorem in the following problems and draw the appropriate conclusion for each of them in the case that you had a Kuhn-Tucker point: I) Min. x 2 + y 2 s. t. : x + y 1 II) Max. x 2 + (y 1) 2 s. t. : x + y 8, 2x + y 4 III) Min. x 2 + (y 1) 2 + 4z 2 s. t. : x + y + 2z 18, 2x + z 4 IV) Min. xy s. t. : x y 0, x 2 + y 2 2 V) Max. x + 4y s. t. : x y 0, x 2 + y 2 2 6. For those problems from the previous exercise for which the hypothesis of the Kuhn-Tucker sufficiency theorem are not fulfilled, 7

analyze the hypothesis of the alternate sufficiency theorem and draw the appropriate conclusion in the case that you had all the Kuhn- Tucker points. 7. Apply the Kuhn-Tucker conditions for the following classical programming problems and solve them: a) Min. x 2 + xy + y 2 x + 2 b) Max. x 3 x 2 + xy y 2 4 c) Max. x 2 + 4x 2y 2 6y d) Min. 3x 2 + xy + 4y 2 s. t. 3x + y = 6 e) Max. 2x + y s. t. x + 2y 2 = 3 f) Min. x 2 + y 2 s. t. x 2 + y = 1 g) Min. x 2 3xy + y 2 + z 2 s. t. x + y + z = 6 8. For the following NLP problem: Max. 6x + 3y x 2 + 4xy 4y 2 s. t. x + y 3 4x + y 9 x, y 0 a) Analyze the constraint qualification and draw the corresponding conclusions. b) State the K-T conditions. c) Study if (2,1) and (0,0) are Kuhn-Tucker points. d) Analyze the hypothesis of the Kuhn-Tucker sufficiency theorem and draw the corresponding conclusions. 9. Given the following NLP problem: Min. 2x + y s. t. xy 4 x y 2 y 0 a) Write the Kuhn-Tucker conditions. b) Prove that the point (-2,0) is a Kuhn-Tucker point. c) Study if (-2,0) is a global minimum. 8

10. For the following problem: Max. (x 2) 2 + (y 2) 2 s. t. x y 8 a) Write the Kuhn-Tucker conditions. x + y 4 x, y 0 b) Check if the points (2,-6) and (8,0) verify said conditions. c) Analyze the K-T sufficient conditions and deduce the corresponding consequence regarding if the previous points are global optima. 11. Given the following problem: Max. 4x 2 2xy y 2 s. t. x 2 + y 2 4 2x + 2y 2 a) Represent the solution set graphically. b) Obtain the only Kuhn-Tucker point knowing that the second constraint must be binding. c) Study if this point is a global maximum. 12. For the following NLP problem: Min. x 1/2 + (y 4) 2 s. t. x + 2y 6 x 2 6x + y 6 y 0 a) Write the Kuhn-Tucker conditions. b) Study if (0,3) is a Kuhn-Tucker point. c) Analyze if the Kuhn-Tucker sufficiency theorem can be applied and draw the corresponding conclusions. 13. Given the following problem: Min. xy s. t. x y 0 x 2 + y 2 2 a) Write the Kuhn-Tucker conditions. b) Check if (0,0) does satisfy these conditions. c) Check if (1,-1) does satisfy them. 9

d) Can you apply the K-T sufficient condition to conclude that they are global minima? e) Explain why it is not possible that both points are global minima. 14. Consider the following problem: Min. x 2 + 3y s. t. x + y 0 x 2 + y 2 2 x 0 a) Write the Kuhn-Tucker conditions. b) Obtain the Kuhn-Tucker point knowing that the first and the second constraints must be binding, but not the third. c) Use the K-T sufficient condition to check that the previous point is a global minimum. 15. Given the following NLP problem: Min. x 2 + 2y 2 + 2xy + 3z s. a. x + 2y + z = 4 x + 3z 6 x 0 a) Can we be sure that the optimal solution (if there is one) will satisfy the Kuhn-Tucker conditions? b) Check that the point (0,3/2,1) satisfies the Kuhn-Tucker conditions and give the value of the associated multipliers. c) Prove that the point (0,3/2,1) is a global minimum of the problem. d) If the right-hand side of the first constraint decreases by 1/6, can you tell, approximately, what would be the new optimal value of the objective function in the new problem? 16. Given the following NLP problem: Min. x 2 + y 2 + 2z 2 yz s. t. x 2y + z = 10 x 2z 8 x 0 a) Can we affirm that the optimal solution of the problem (if there is one) will satisfy the Kuhn-Tucker conditions? Justify your answer. 10

b) Write the Kuhn-Tucker conditions. c) Check if the points (2,-4,0) and (6,-1,2) satisfy the Kuhn-Tucker conditions and, if they do, calculate the values of the multipliers. d) What can you say about the optimality of the points (2,-4,0) and (6,- 1,2)? e) This problem is bounded, if the right-hand side term of the first constraint changes to 10.25, give the approximate optimal value of the new problem. 17. Given the following NLP problem: Min. x 2 + y 2 + 2z 2 yz s. t. x 2y + z 10 x 2z = 2 y 0 a) Can we affirm that the optimal solution of the problem (if there is one) will satisfy the Kuhn-Tucker conditions? Justify your answer. b) Write the Kuhn-Tucker conditions. c) Check if the points (2,-4,0) and (6,-1,2) satisfy the Kuhn-Tucker conditions and, if they do, calculate the values of the multipliers. d) What can you say about the optimality of the points (2,-4,0) and (6,- 1,2)? e) This problem is bounded, if the right-hand side term of the first constraint changes to 9.75, give the approximate optimal value of the new problem. 18. Given the following consumer problem: Max. U(x, y) = x 1/2 + y 1/2 s. t. 2x + y 12 x 1, y 1 a) Write the Kuhn-Tucker conditions. b) Check if (2, 8) is a Kuhn-Tucker point. c) Study if (2, 8) is the global maximum of the consumer problem. 19. Given the following problem that minimizes the costs in a firm: Min. C(x, y, z) = 6x + 4y + z s. t. x + y + z 1/2 92 x + 2y 100 a) Write the Kuhn-Tucker conditions. 11

b) Obtain the only Kuhn-Tucker point of the problem, knowing that z = 16. c) Study if this point is the global minimum of the problem. d) If the minimum production required in the first constraint changes from 92 to 90, calculate, approximately, the variation in the optimal costs. 20. The cost function in a firm that designs x and maintains y web pages is given by: C(x, y) = x 3 + 3y 2 6xy + 32x + 12y The firm spends exactly 89 weekly hours to the design and maintenance of web pages. Each web page designed needs 7 weekly hours, and the maintenance of one web page requires 15 weekly hours. Currently, 2 web pages are designed and 5 are maintained each week. a) Reason out if the firm is minimizing its costs. Is it a global minimum? Under which economical conditions can you guarantee that the minimum obtained is global? b) Reason out (without solving the problem again) what the optimal cost would approximately be if the firm spent exactly 90 hours designing and maintaining web pages. 21. Consider the following problem: Max. B = 300x + 200y x 2 s. t. 2x + y 20 x + 2y 5 x, y 0 where B is the benefits function of a firm and variables x, y represent the amounts of two articles A and B produced. a) Can we affirm that the optimal solution (which we still have not obtained) satisfies the Kuhn-Tucker conditions? b) Study if the solution (x,y) = (5,0) satisfy said conditions. c) State the K-T sufficient condition for maximization problems, and use it to prove that the previous point is a global optimum of the problem. d) Reason out what would happen to the benefits for each unit of product B that was produced. 22. Given the basic consumer problem: Max. U(x, y) s. t. p 1 x + p 2 y M x, y 0 12

a) Analyze the problem from the point of view of non-linear programming: existence of solution, constraint qualification, and applicability of the local-global theorem and the Kuhn-Tucker sufficient condition. b) Write the Kuhn-Tucker conditions. c) Prove that if the consumer is rational, i.e. he prefers more over less (marginal utilities strictly positive), the resolution of the Kuhn-Tucker conditions will imply that the optimal solution spends all the rent. d) Solve the problem when U(x, y) = 10 ln(x) + 5ln (y), M=180 prices are p 1 =5 and p 2 =3. 23. Check the boxes corresponding to true statements: Knowing that a certain point is regular and Kuhn-Tucker is enough to affirm that it is a global optimum. Knowing that a certain point is regular and no Kuhn-Tucker is enough to dismiss it as a global optimum. Knowing that a certain point is not regular is enough to dismiss it as a global optimum. Knowing that a certain point is not Kuhn-Tucker is enough to dismiss it as a global optimum. Knowing that a problem satisfies the constraint qualification is enough to state that only a Kuhn-Tucker point can be a global optimum. Knowing that a problem satisfies the constraint qualification is enough to affirm that the best Kuhn-Tucker point is a global optimum. Knowing that a problem does not satisfy Weierstrass theorem is enough to affirm that no Kuhn-Tucker point is a global optimum. Knowing that a problem satisfies the hypothesis of the K-T sufficient conditions is enough to affirm that every Kuhn-Tucker point is a global optimum. Knowing that a problem does not satisfy the hypothesis of the K-T sufficient conditions is enough to affirm that no Kuhn-Tucker point is a global optimum. 13

24. Join each statement of the first column with the right consequence(s) of the second column: Constraint qualification is satisfied The problem has a global optimum Weierstrass theorem applies S is bounded The Kuhn-Tucker sufficiency theorem is fulfilled The global optimum is a K-T point S is convex Every K-T point is a global optimum 14

UNIT 3.- AN INTRODUCTION TO LINEAR PROGRAMMING 1. Solve the following problems graphically, obtain the optimal solution, and say of what type it is. Find out if the solution set is bounded or not. a) Max. x + y s. t. x + y 2 x + 2y 6 2x + y 6 x, y 0 b) Max. 2x + y s. t. x + y 2 x + 2y 6 2x + y 6 x, y 0 c) Max. x + y s. t. x + y 2 y 4 x, y 0 d) Min. x + y s. t. x + y 2 y 4 x, y 0 e) Max. x + y s. t. x + y 2 y 4 x, y 0 f) Max. y s. t. x + y 2 y 4 x, y 0 2. Reason out if it each of the following situation is possible. For those that are possible, give a linear problem illustrating it. a) An unbounded linear problem with a bounded solution set. b) An unbounded linear problem with an unbounded solution set. c) A linear problem with optimal solution(s) and a bounded solution set. d) A linear problem with optimal solution(s) and an unbounded solution set. 3. Write the following problems in standard and canonical form: a) Max. x + y s. t. x + y = 2 x + 2y 6 2x + y 6 x 0, y 0 b) Max. 2x + 3y + z s. t. 4x + 3y + z 20 x + y 20 x 0, y 0, z free c) Min. x + y s. t. x + y 2 What is the maximum number of basic or non-zero variables that a basic feasible solution of each of these problems can have? 4. Given the following LP problem: Max. 2x + 3y + z 15

s. t. x + 2y + z 30 x + y 20 x, y, z 0 Reason out which of the following points is/are BFS. a) (10,10,0,0,0) c) (20,0,5,5,0) e) (20,0,5,0,0) b) (0,0,0,30,20) d) (0,0,30, 0,20) 5. Given the following LP problem, calculate all its basic feasible solutions and say the basis associated with each of them. Max. 2x 3y s. t. x + y 1 x y 0 x, y 0 6. Given the following linear programming problem: Max. 2x + 4y + 3z s. t. x + y 8 x + 3y + z 20 2x z = 15 x 0, y 0 z free a) Transform the problem into standard form and write the technical matrix A. b) Obtain any of its basic feasible solutions. 7. Given the following problem: Max. x 1 + 2x 2 + 4x 3 s. t. x 1 + x 2 10 x 1 + 2x 2 + x 3 = 14 x 1, x 2, x 3 0 a) Determine the basic feasible solution associated with x B =(x 1, x 2 ). b) Obtain two more basic feasible solutions. c) Obtain a feasible solution that is not basic. 8. Consider the following linear programming problem: Max. f(x, y, z) s. t. x + y + z 6 x + 2y z = 6 16

Obtain: x, y, z 0 a) A basic feasible solution that is not degenerate, giving its associated basis. b) A degenerate basic feasible solution, giving its associated basis. 9. Given the following linear programming problem: Max. 3x + y s. t. y 4 6x + 2y 12 x, y 0 a) Draw the set of feasible solutions. Identify all the basic feasible solutions of this problem graphically and give the value of the variables (x, y, s 1, s 2 ) in each of them. b) Solve the problem. 10. A given linear problem has the following solution set: 5 4 3 2 1 1 2 3 4 5 6 a) Enumerate all the BFS of this problem. b) If the objective function of said problem is known to be 4x-y, can the optimal solution of the problem be calculated? If the answer is yes, say which one it is. If the answer is no, say what should occur for this to be possible. c) Answer the question b) but considering the objective function x+y. d) 11. A maximization linear problem in standard form has four basic feasible solutions, which we will call BFS 1, BFS 2, BFS 3, and BFS4. The objective function for said basic feasible solutions takes values: z 1 = 3, z 2 = 4, z 3 = 8, z 4 = 6, respectively. Explain if we can or cannot tell that BFS 3 is the optimal solution of the problem. 12. Consider the problem Max. 4x + 2y + 4z s. t. x 5 2x + y + z = 4 17

y + 2z = 3 x, y, z 0 a) Determine if there is a basic feasible solution with the basic variables x, y, z. b) Check if (1/2, 3, 0, 9/2), (1, 1, 1, 4), (0, 5, 1, 5) are basic feasible solutions. c) Obtain all the basic feasible solutions and the value of the objective function for each of them. d) Justify that the problem has optimal solution and calculate it. 13. The following figure represents the solution set of a linear programming problem in canonical form (located in the quadrant x > 0, y > 0). a) How many basic variables does a basic feasible solution have? b) Which of the black points are basic feasible solutions and which not? c) Say which of the three points can be optimal solutions of the problem and which not. d) Can x be a non-basic variable in some solution? e) Can the problem be unbound? Can it be infeasible? Can it have edge solutions? And infinite edge solutions? 14. Let F(x,y,z) = x+y+2z be the objective function of a linear problem that we want to maximize. All three variables are nonnegative. next. a) Add one constraint so that the problem has only one (vertex) optimal solution. Obtain all the basic feasible solutions and the value of the objective function for each of them. b) Add one constraint so that the problem has multiple (edge) solutions. Obtain all the basic feasible solutions and the value of the objective function for each of them. c) Add one constraint so that the problem is unbounded. d) Add one constraint so that the problem is infeasible. Note: the constraints are no accumulated from one question to the 18

UNIT 4.- THE SIMPLEX METHOD 1. Given the followng simplex tableau corresponfing to a maximization problem: 1 6 2 1 2 1 x 1 x 2 x 3 x 4 x 5 x 6 1 x 1 1 1 0 0 4 1 2 2 x 3 0 1 1 0 2 2 1 1 x 4 0-2 0 1 5 0 1 z j 1 1 2 1 13 5 5 w j 0 5 0 0-11 -4 a) Which variables are basic and which non-basic? b) What is the solution associated with this tableau? What is the value of the objective function for this solution? Is it a basic feasible solution? c) Check that this tableau satisfies the conditions that every simplex tableau must meet. d) What effect would it have on the objective function if any of the nonbasic variables entered the basis? e) Reason out that the tableau is no optimal. Point out which variable must enter, which must leave, and the pivot element. f) Calculate the next tableau. Chech that the new tableau satisfies all the conditions that a simplex tableau must meet. 2. Answer the questions from the previous exercise for the following simplex tableau corresponding to a maximization problem: 5 1 2 0 0 x y z s 1 s 2 5 x 1 0-1/3 2/9 5/9 0 1 y 0 1 1/3 1/9-2/9 2 z j 5 1 4/3 11/9 23/9 w j 0 0 2/3-11/9-23/9 2 3. Given the following simplex tableaus corresponding to maximization problems, say if they are optimal and, if they are, say which type of optimal solution we have. Point out, when possible, the optimal solution given in the tableau. If there are more optimal solutions, otain them. 4 4 6 0 0 x y z s 1 s 2 4 x 1 4 2 1 0 5 0 s 2 0-6 -5-2 1 10 z j 4 16 8 4 0 w j 0-12 -2-4 0 20 19

4-5 3 0 0 x y z s 1 s 2 3 z 1/3-2/3 1 1/3 0 2 0 s 2 6-5 0 2 1 24 z j 1-2 3 1 0 w j 0-3 0-1 0 6-1 1 0 0 x y s 1 s 2-1 x 1 0-1/2-1/2 1/2 1 y 0 1 1/2-1/2 1/2 z j -1 1 1 0 w j 0 0-1 0 0 4 8 1 0 0 x y z s 1 s 2 0 s 1-3/2 0 5/2 1 0 10 8 y -1/2 1 1/2 0 0 1 0 s 2-1/2 0-1/2 0 1 0 z j -4 8 4 0 0 w j 8 0-3 0 0 8 4. For the following maximization problems, an associated simplex tableau is given: Max 2x 1 +4x 2 +8x 3 s.t.: 2x 1 +2x 2 +4x 3 8 2x 1 +x 2 +2x 3 6 x 1,x 2,x 3 0 2 4 8 0 0 x 1 x 2 x 3 s 1 s 2 4 x 2 1 1 2 1/2 0 4 0 s 2 1 0 0-1/2 1 2 z J 4 4 8 2 0 1 w j -2 0 0-2 0 6 Max 2x+3y s.t.: x+y+z = 4 y+z 2 x,y,z 0 2 3 0 0 x y z s 2 x 1 0 0-1 2 0 z 0 1 1 1 2 z J 2 0 0-2 w J 0 3 0 2 4 20

Max -3x-4y s.t.: 2x+3y 10 x+3y 5 x,y 0-3 -4 0 0 x 1 x 2 s 0 s 1 1 0 1 1 5-4 x 2 1/3 1 0-1/3 5/3 z J -4/3-4 0 4/3 - w J -5/3 0 0-4/3 20/3 1 s 2 Answer for each of the previous tableaus: Is it the optimal tableau? Give the value of the variables and the objective function for the corresponding basic feasible solution. Point out which variables are basic or non-basic in the given solution. Give the associated basic matrix and calculate its inverse matrix. Can this inverse matrix be found in the tableau? 5. Given the following LP problem: Max. 2x + 2y + 10 s. t. 4x + 2y 16 x + 2y 8 x, y 0 With how many different tableaus can the simplex algorithm begin? Write two of them and determine, if appropriated, the entering and the leaving variables and the pivot at each of them. 6. Given the following LP problem: Max x 1 +3x 2 s.t.: x 1 +x 2 6 -x 1 +2x 2 8 x 1,x 2 0 a) Solve it using the simplex method. b) Draw the solution set of the problem. Single out the basic feasible solutions corresponding to each tableau in the graph. 7. Solve this problem with the simplex method: Min. x 1 + 2x 2 + 4x 3 s. t. x 1 + x 2 10 x 1 + 2x 2 + x 3 = 14 x 1 0, x 2 0, x 3 free 21

8. Given the following LP problem: Max. 2x 1 + x 2 s. t. x 1 + x 2 2 x 1 + 2x 2 6 2x 1 + x 2 6 x 1 0, x 2 0 and knowing that the optimal solution is x 1 =2 and x 2 =2, obtain the optimal simplex tableau without doing any iteration. What type of solution is it? 9. Given the following LP problem Max x 1 +x 2 s.t.: -x 1 +x 2 2 x 2 4 x 1,x 2 0 Obtain, without performing any iteration, the simplex tableau with the basic variables x 2 and s 2. Is it the optimal tableau? If so, give the solution and the value of the objective function. Otherwise do one more iteration. 10. Given the following LP problem: Max 2x 1 -x 2 +x 3 s.t.: x 1 +x 2 +x 3 6 -x 1 +2x 2 4 x 1,x 2,x 3 0 and knowing that in the optimal solution x 1 =6 and x 2 = x 3 =0, calculate the optimal simplex tableau without performing any iteration. What type of solution is it? 11. Given the following linear programming problem: Min. x + 4y s. t. x + 3y = 4 1 y 8 x 0 Show that the point (x, y +, y -, s 1, s 2 ) = (7,0,1,9,0) is a basic feasible solution of the problem in standard form and obtain the associated simplex tableau. Study if this tableau is optimal. If it is, write the solution of the original linear problem, if not, say which variable should enter, which one should leave and the pivot element of the next iteration. 22

12. Consider the following problem: Max. 5x + 2y s. t. x + y 3 x + y = 4 2x + y 2 x, y 0 a) Write the corresponding artificial problem with the minimum possible number of artificial variables. b) Solve the problem using the big M method. c) Is it necessary to add artificial variables to solve the problem? If it is not necessary, give a possible initial simplex tableau. 13. Consider the following linear problem: Max 3x+4y+10z s.t.: 2x+y +z = 1 2x-z = 2 2x-2z 2 x,y,z 0 a) Write the corresponding artificial problem with the minimum possible number of artificial variables. b) Obtain the first simplex tableau associated with the problem in a) and do one iteration of the simplex method. Study if the obtained tableau is optimal. If it is, say which type of solution the original problem has. Otherwise say which variables must enter and leave the basis. c) How could we have solved the original problem by means of the simplex method without using the big M method? 14. Consider the following linear problem: Max -20x-30y-16z s.t.: 5/2x+3y +z 3 -x-3y-2z -4 x,y,z 0 a) Write the corresponding artificial problem with the minimum possible number of artificial variables. b) Obtain the first simplex tableau associated with the problem in a). 23

15. The optimal tableau of the simplex method corresponding to a certain LP maximization problem in canonical form is: 3 3 0 0 x y s 1 s 2 3 x 1 0-1/3 4/3 4 3 y 0 1 1/3-1/3 1 z j 3 3 0 3 w j 0 0 0-3 15 a) Is there a solution? Is it the only one? If it is the only optimal solution, give its values. It there are more optimal solutions, obtain the remaining ones. b) Calculate matrix A and the right-hand side terms vector b. Write the original problem. 16. What values in the following tableau must be wrong? x y s 1 s 2 s 3 s 1 1/3 0 1-1 1-2 y 0 2 0-1/3 2/3 2 x 1 0 0 2/3-1/3 2 Z j 1 2 0 0 1 W j 0 1 0 0-1 6 17. We are solving a maximization linear problem by means of the simplex method. Explain which conclusion we can draw in each of the following cases: a) There is a non-basic variable with w j >0, but no basic variable satisfies the leaving choice rule. b) The tableau is optimal and there is a non-basic variable with zero reduced cost. c) The tableau is optimal and all the non-basic variables have reduced cost strictly less than zero. 24

18. Give an example of a linear problem having at least one equality constraint that does not need artificial variables in order for the basic matrix of the first tableau to be the identity matrix 19. When, in a linear programming problem, we obtain a simplex tableau where all the w j 0 and an artificial variable is basic, what do we know about the problem? 20. Let us consider a maximization linear problem. Answer the following questions: a) What sign must the values of the basic variables have in any simplex tableau? b) What sign must the value of the objective function have in any simplex tableau? c) Are there any variables for which the value of w j is known a priori for any given tableau? d) Will we always be able to construct at least one simplex tableau? e) When will we be able to find the inverse basic matrix (B -1 ) associated with a certain tableau in the talbeau itself? Where will it be? f) When will the first box of the simplex tableau contain the identity matrix? Where will it be? g) When will the first box of the simplex tableau be the matrix A? 25

UNIT 5.- DUALITY IN LINEAR PROGRAMMING 1. Formulate the associated dual problem of: Problem 1 Problem 2 Max. x 1 + x 2 + x 4 s. t. 2x 1 + x 2 + 4x 3 = 10 x 1 + 2x 4 8 x i 0, i = 1,3, x 4 free, x 2 0 Min. x 1 + x 2 + x 4 s. t. 2x 1 + x 2 + 4x 3 10 x 1 + 2x 4 = 8 x i 0, i = 1,2, x 4 free, x 3 0 Problem 3 Max. F(x) = x 1 + 2x 2 + x 3 + 3x 4 s. t. 4x 1 + 2x 2 + 3x 3 + 2x 4 20 2x 1 + x 2 + x 3 + x 4 16 x 1 2x 2 + 2x 3 + 4x 4 = 10 x 1 0, x 2 0, x 3 free, x 4 0 Problem 4 Min. F(x) = x 1 + 2x 2 + x 3 + 3x 4 s. t. 4x 1 + 2x 2 + 3x 3 + 2x 4 = 20 2x 1 + x 2 + x 3 + x 4 16 x 1 2x 2 + 2x 3 + 4x 4 10 x 1 0, x 2 0, x 3 0, x 4 free Problem 5 Max. 10x 1 + x 2 s. t. 4x 1 + x 2 8 x 1 + x 2 5 x 1 0, x 2 free Problem 6 Max. 4x 1 + x 2 s. t. 2x 1 + x 2 8 x 1 + 2x 2 = 6 x 1 free, x 2 0 Problem 7 Max. 5x + 2y s. t. x + y 3 x + y = 4 2x + y 2 x, y 0 Problem 8 Max. 5x + y s. t. 2x y 3 x 5y = 1 2x y 6 x 0, y 0 2. Solve the following problems graphically: a) Min. 2x + 3y + 5s + 2t + 3u s. t. x + y + 2s + t + u 4 2x + 2y + 3s + t + u 3 x, y, s, t, u 0 26

b) Max. 4x 1 + 3x 2 + x 3 s. t. x 1 + x 2 + x 3 4 x 1 + x 3 4 x i 0, i 3. Formulate the dual problem of the following linear problem: Max 2x y s. t. : x y 1 x + y 1 x 1 Can anything be said about the optimal solution of the dual problem without using the simplex method? 4. Formulate and solve the dual problem associated with the following linear programming problem: Min. x + 4y s. t. x + 3y = 4 1 y 8 x 0 Solve the dual problem without using the simplex method. 5. Write a linear programming problem whose dual problem is infeasible. Justify your answer without writing the dual problem. 6. We know that (0,6,0) is the optimal solution of Max. x 1 + 4x 2 + x 3 s. t. x 2 + 2x 3 4 x 1 + 2x 2 + 4x 3 = 12 x 1 0, x 2 0, x 3 0 Formulate the dual problem and obtain its optimal solution (main variables, slack variables and objective function). 7. We know that (1,6,0) is the optimal solution of Min. 8x 1 + 3x 2 + 2x 3 s. t. x 2 + 2x 3 6 2x 1 + x 2 + 4x 3 = 8 x 1 0, x 2 0, x 3 0 27

Formulate the dual problem and obtain its optimal solution (main variables, slack variables and objective function). 8. We know that (2,1,0) is the optimal solution of Min. 3x 2y + 5z s. t. 3x + 2y 6 x + 5z = 2 x + 3y + 2z 5 x 0, z 0 Formulate the dual problem and obtain its optimal solution (main variables, slack variables and objective function). 9. We know that (0,11/3,-2/3) is the optimal solution of Max. 6x + 2y + 5z s. t. 3x + y + z 3 2x + 3z = 2 5y + 2z 5 x 0, z 0 Formulate the dual problem and obtain its optimal solution (main variables, slack variables and objective function). 10. We know that (0,4,-4) is the optimal solution of Min. 3x 1 + x 2 x 3 s. t. 4x 1 + 2x 2 x 3 = 12 x 1 2x 3 8 x 1 0, x 2 0, x 3 0 Formulate the dual problem and obtain its optimal solution (main variables, slack variables and objective function). 11. Given a maximization linear problem in canonical form whose solution is: (x 1, x 2, x 3, s 1, s 2 )=(1,0,3,0,0) F(1,0,3)=20 (w x1, w x2, w x3, w s1, w s2 )=(0,-2,0,-2,-4) Say which variables are basic and which are nonbasic in the dual optimum, as well as their value, the dual w j and the value of the objective function for the optimal solution. 28

12. For the following problems: a) Formulate the dual problem. b) Obtain the complete dual optimal solution (main and slack variables, reduced costs and objective function). c) Say what type the dual optimal solution is (degenerate or not degenerate; vertex, edge or infinite edge). Problem 1 Max. F(x) = 2x 1 + 4x 2 + 8x 3 s. t. 2x 1 + 2x 2 + 4x 3 8 2x 1 + x 2 + 2x 3 6 x 1 0, x 2 0, x 3 0 Optimal tableau of problem 1 2 4 8 0 0 x 1 x 2 x 3 s 1 s 2 4 x 2 1 1 2 1/2 0 4 0 s 2 1 0 0-1/2 1 2 Z J 4 4 8 2 0 W j -2 0 0-2 0 16 Problem 2 Max. 2x + 6y s. t. x + 3y 9 2x + y 8 x, y 0 Optimal tableau of problem 2 2 6 0 0 x y s 1 s 2 2 x 1 0-1/5 3/5 3 6 y 0 1 2/5-1/5 2 Z j 2 6 2 0 W j 0 0-2 0 18 29

Problem 3 Max. x + y 5z s. t.. 2x y z 12 x + 2y 3z 6 x, y, z 0 Optimal tableau of problem 3 1 1-5 0 0 x y z s t 1 x 1 2-3 0 1 6 0 s 0-5 5 1-2 0 Z j 1 2-3 0 1 W j 0-1 -2 0-1 6 Problem 4 Max. x + 2y s. t. x y 2 2x + y 6 x + 2y 6 x, y 0 Optimal tableau of problem 3 1 2 0 0 0 x y s 1 s 2 s 3 0 s 1 0 0 1-1 1 2 2 y 0 1 0-1/3 2/3 2 1 x 1 0 0 2/3-1/3 2 Z j 1 2 0 0 1 W j 0 0 0 0-1 6 Problem 4 Max. x 5y + 3z s. t. x 2y + 3z 6 4x y 6z 12 x, y, z 0 30

Optimal tableau of problem 4 1-5 3 0 0 x y z s t 3 z 1/3-2/3 1 1/3 0 2 0 t 6-5 0 2 1 24 Z j 1-2 3 1 0 W j 0-3 0-1 0 6 13. Let us consider the following dual problem (DP) of a certain primal problem (PP) Min 12y 1 + 6y 2 + 27y 3 s. t. : 3y 1 + y 2 + 5y 3 3 y 1 + y 2 + 3y 3 2 y 1, y 2, y 3 0 with dual optimal solution (1/2, 3/2, 0, 0, 0). Write the primal problem and obtain its optimal solution from the dual optimal solution. 14. A firm produces two articles in daily amounts x and y. There are 200 daily hours of labour available, and the second article requires a raw material of which there are 30 daily units available. The unitary profits are 2 and 5 m.u. respectively. The problem is Max. 2x + 5y s. t. 3x + 5y 200 3y 30 x, y 0 Knowing that the optimal solution is (50,10), calculate the main dual variables and explain their meaning. Would the firm be interested in buying 10 extra hours at a price of 1 m.u. each? 15. The solution set of a certain linear problem is S = {(x, y) R 2 / 2x + 5y 10, 2x + y 6, x + 3y 3, x + 2y 2, x 0, y 0} Let us suppose that we solve the associated dual problem. Analyze which of the following cases are possible: a) The dual problem has optimal solution. b) The dual problem is infeasible. c) The dual problem is unbounded. 31

16. Given a linear problem with five variables, it is known that (0,3,0,1,4) is a feasible solution. Let us suppose that we solve the associated dual problem. Analyze which of the following cases are possible: a) The dual problem has optimal solution. b) The dual problem is infeasible. c) The dual problem is unbounded. 17. Given a minimization linear problem in canonical form, it is known that the optimal solution is degenerate. What can we know about the dual optimal solution? 18. Is it possible to solve a linear problem with 4 variables and 2 constraints graphically? If so, explain how; else, give the maximum number of variables and constraints that a problem should have so that it can be solve graphically. 19. Among the three possible types of solution of a linear problem (infeasible, bounded, and unbounded), say which types of solutions can appear in the dual problem of a linear problem for each the following cases: a) The primal problem is infeasible. b) The primal problem is bounded. c) The primal problem is unbounded. 20. Let us consider the following LP problem: (P) And let x 0 S / f(x 0 ) = 44. Min f(x) s. t. : x S Let problem (D) Max g(μ) be its dual problem and μ 0 T / g(μ 0 ) = 30 s. t. : μ T Reason out if the following situations can occur: a) P is unbounded b) D is infeasible c) D is unbounded d) x 0 is an optimal solution of P and µ 0 is an optimal solution of D e) P has optimal solution but x 0 is no optimal solution of P f) There is a solution μ 0 of problem D with g(μ 0) = 47 g) The optimal value of the objective function os P is 30 h) The optimal value of the objective function of P is 30 i) The optimal value of the objective function of D is 44 32

21. Given the following problem: Min x + 2y s. t. : x 9 x + y 13 x, y 0 a) Solve it using the simplex method using as basic variables for the initial tableau x B =(x,y) b) If possible, use the optimal tableau of the problem to get the optimal solution of the dual problem (main and slack variables, reduced costs, and objective function value). c) Check that the formula (λ,µ)=c B B -1 leads to the same optimal solution. d) Obtain the optimal tableau of the dual problem and check that the value of all the variables and the reduced costs of the optimal solution coincide with those obtained in b). 33

UNIT 6.- SENSITIVITY ANALYSIS AND POST- OPTIMIZATION 1. If the coefficient of a non-basic variable in the objective function changes, but without leaving its sensitivity interval, analyze the changes that the composition of the basis, the value of the basic variables, and the value of the objective function suffer for the optimal solution of the new problem. 2. Given the following linear programming problem: Max. F(x) = 2x 1 + 4x 2 + 8x 3 s. t. 2x 1 + 2x 2 + 4x 3 8 2x 1 + x 2 + 2x 3 6 x 1 0, x 2 0, x 3 0 Whose optimal solution is given in the following tableau: 2 4 8 0 0 x 1 x 2 x 3 s 1 s 2 4 x 2 1 1 2 1/2 0 4 0 s 2 1 0 0-1/2 1 2 Z J 4 4 8 2 0 W j -2 0 0-2 0 16 a) Obtain the sensitivity interval of c 1 from the optimal tableau. b) Get the new optimal solution when c 1 =5. 3. If the coefficient of a basic variable in the objective function changes, but without leaving its sensitivity interval, analyze the changes that the composition of the basis, the value of the basic variables, and the value of the objective function suffer for the optimal solution of the new problem. 4. Given the following problem: whose optimal tableau is: Max. 2x + 6y s. t. x + 3y 9 2x + y 8 x, y 0 34

2 6 0 0 x y s 1 s 2 2 x 1 0-1/5 3/5 3 6 y 0 1 2/5-1/5 2 Z j 2 6 2 0 W j 0 0-2 0 18 a) Obtain the sensitivity interval of the coefficient of variable x in the objective function and explain its meaning. b) If we set c 1 = 10, is the tableau still optimal? What is the optimal value of the objective function in this case? 5. If the right-hand side term of a constraint changes, but without leaving its sensitivity interval, analyze the changes that the composition of the basis, the value of the basic variables, and the value of the objective function suffer for the optimal solution of the new problem. 6. For the following LP problem: Max. 2x 1 + x 2 + 3x 3 s. t. x 1 + 2x 3 12 3x 1 + 2x 2 + x 3 18 x 1, x 2, x 3 0 a) Obtain the optimal simplex tableau starting from the following one, which you have to complete: 2 1 3 0 0 x 1 x 2 x 3 s 1 s 2 1 x 2 5/4 1 0 1/4 1/2 3 x 3 1/2 0 1-1/2 0 z j w j x B b) Perform the sensitivity analysis of b 2 and explain the meaning of the interval. 7. Given the following linear programming problem: Min. x + 5y 3z s. t. x 2y + 3z 6 4x y 6z 12 x, y, z 0 35

And the following optimal tableau corresponding to the equivalent maximization problem: 1-5 3 0 0 x y z s t 3 z 1/3-2/3 1 1/3 0 2 0 t 6-5 0 2 1 24 Z j 1-2 3 1 0 W j 0-3 0-1 0 6 a) If the right-hand side of the first constraint changed to 12 (b 1 =12), what would the optimal solution of the problem be? Write the value of the variables and the objective function and argue what type of solution it is. 8. If a technical coefficient associated with a nonbasic variable changes, but without leaving its sensitivity interval, analyze the changes that the composition of the basis, the value of the basic variables, and the value of the objective function suffer for the optimal solution of the new problem. 9. Given the following linear problem: Max. 3x 1 + x 2 + 3x 3 + 4x 4 s. t. 2x 1 + 2x 2 + 3x 3 + 8x 4 20 3x 1 + 5x 2 + 2x 3 + 2x 4 15 x i 0, i = 1,...,4 with the following optimal tableau: 3 1 3 4 0 0 x 1 x 2 x 3 x 4 s 1 s 2 3 x 1 1 11/5 0-2 -2/5 3/5 1 3 x 3 0-4/5 1 4 3/5-2/5 6 Z j 3 21/5 3 6 3/5 3/5 W j 0-16/5 0-2 -3/5-3/5 21 a) If the coefficient of x 4 in the first constraint changes to a 14 =3, calculate the optimal solution. b) Obtain the sensitivity interval of and explain its a 14 meaning. 36

10. If we add a new non-basic variable to the problem, analyze how it will affect to the composition of the base, the value of the basic variables, and the value of the objective function in the optimal solution of the new problem. 11. For the following LP problem: Max. x 1 + 4x 2 + x 3 s. t. x 2 + 2x 3 4 x 1 + 2x 2 + 4x 3 = 12 x 1 0, x 2 0, x 3 0 a) Obtain the optimal simplex tableau without doing any iteration, knowing that the optimal solution is (x 1, x 2, x 3 )=(0,6,0). b) If a new variable x 4 with coefficient c 4 =3 in the objective function and technical coefficients A 4 = 2 is added, reason out if the optimal 1 solution in a) is still optimal and, if it is not, obtain the new optimal solution. 12. If we add a new constraint to the problem, analyze how it will affect to the composition of the base, the value of the basic variables, and the value of the objective function in the optimal solution of the new problem. 13. Given the following linear programming problem: Min. 8x + 3y + 2z s. t. y + 2z 6 2x + y + 4z 8 x, y, z 0 with the following optimal tableau corresponding to the equivalent maximization problem: -8-3 -2 0 0 x y z s t -2 z 0 1/2 1-1/2 0 3 0 t -2 1 0-2 1 4 Z j 0-1 -2 1 0 W j -8-2 0-1 0-6 Calculate the optimal solution if we add the constraint x + y + z 2. Which variables are basic? 37

14. Consider the following LP: Max. x 1 + 2x 2 s. t. x 1 + x 2 4 2x 1 + x 2 6 x 1, x 2 0 Knowing that the basic variables in the optimal solution are s 2 and x 2 : a) Obtain the optimal solution. b) Perform the sensitivity analysis for: c 1, c 2, b 1, b 2, and a 11. c) Obtain, starting from the known optimal solution, the optimal solution for each of the following changes: c 1 =3/2, c 1 =3, c 2 =1/2, c 2 =4, b 1 =5, b 1 =10, b 2 =8, a 11 =2, and a 21 =2. d) Obtain, starting from the known optimal solution, the optimal solution if a new variable x 3 with c 3 =6 and Pt 3 = (2,2) is added. Do the same if a new constraint x 1 + 2x 2 8 is added. e) Starting from the initial optimal solution, obtain the value that the coefficient a new variable should have in the objective function in order for it to enter the basis, if its coefficients in the constraints are a 13 =2, and a 23 =1. (Note: each question of this problem must be solved considering the original problem, and not the results of the previous question.) 15. Consider the problem: Max. 2x + 3y s. t. 2x + y 4 3x + y 4 x, y 0 Its optimal talbeau is: 2 3 0 0 x y s 1 s 2 3 y 2 1 1 0 4 0 s 2 1 0-1 1 0 Z j 6 3 3 0 W j -4 0-3 0 12 38

a) Obtain the optimal solution and the value of the objective function in the optimal solution if the following values are changed simultaneously: c 1 =5, c 2 =2. Which variables are now basic? Why are they different from the current ones if both coefficients lie in the sensitivity interval? b) Do as above if c 1 =5, b 1 =6. Which variables are now basic? c) Do as above if c 1 =5, a 11 =1. Which variables are now basic? 16. A bottler company bottles and markets three types of wine A, B, and C, obtaining a profit from each barrel of 50, 25, and 20 m.u., respectively. Each barrel goes through two phases, filling and sealing. There are 640 hours of labour available each week for the first process, and 900 hours for the second one. The number of hours that one barrel must spend at each phase is given in the following table: Filling Sealing A 16 30 B 4 5 C 6 10 The optimal solution of the linear model that represents this problem is (0,160,0,0,100). a) How many barrels of type C would you fill if the total number of hours available at the filling section were 700? b) If the profit for a barrel of type A was 55 m.u., how many barrels of type B would be filled? c) How much ca the unitary profit of the barrels of type A decrease without affecting the solution? d) Obtain the sensitivity interval for the profit of a barrel of type B and explain its meaning. 17. A company is producing two articles in amounts x and y, whose maximum demand (for both products jointly) is estimated to be 200 units. The unitary production cost is 2 m.u. for the first product and 3 m.u. for the second one. The company s budget is 500 m.u.. Furthermore, each unit of the first article yields 3 m.u. of benefit, while the second article yields 1 m.u. per unit. a) Obtain the production that maximizes the benefits, as well as the maximum benefit. b) Study the effect of increasing the unitary benefit of the second article from 1 to 2 monetary units. 39

c) Do as above with a unitary benefit of 4 m.u.. d) Calculate the sensitivity interval of the unitary benefit of each article. e) Calculate the sensitivity interval of the unitary cost of the second article. f) Calculate the sensitivity interval of the demand. g) Study how an increase of the demand up to 250 units would affect the solution. h) Do as above with a demand of 300 units. i) Let us suppose that the company is forced to produce at least half as many units of the second article than of the first one, that is, y x/2 or, equivalently, x 2y 0. Obtain the new optimal solution. 18. A company is considering the following linear problem: Max. 3x 1 + x 2 + 2x 3 (benefits) s. t. 2x 1 + x 2 + x 3 18 (capital) 3x 1 + 2x 2 + 3x 3 15 (hours of labor) x 1, x 2 x 3 0 a) Calculate the optimal simplex tableau knowing that the optimal point is (5, 0, 0). b) Perform the sensitivity analysis of a 12 and a 22. c) What effect would it have to reduce the hours of labor from 15 to 12? d) Calculate the value of the dual variables. Argue if the company would be more interested in having more capital or more labor hours in order to increase its benefits. 40

UNIT 7.- INTEGER LINEAR PROGRAMMING 1. Given a certain integer liner programming problem with the following branchand-bound tree: F(x,y,z)=41,36 x=0.05 y=8 z=8.58x<=0 0 x>=1 1 2 F(x,y,z)=41.33 x=0 y=8 z<=8 z=8.67 z>=9 1.1 Infeasible 1.2 F(x,y,z)=39 x=0 y=7.67 z=8 y<=7 y>=8 F(x,y,z)=37 x=0 y=7 z=8 1.2.1 1.2.2 infrasible F(x,y,z)=26.4 x=1 y=4.4 z=4.6 a) Reason out if the objective of the problem is to maximize or minimize. b) Have we reached the optimal solution? If we have, explain why, else say from which node we should branch and with which constraints. c) Write the problem solved in node 1.2.1. 2. Given a certain integer linear programming problem with the following branch-and-bound tree: 0 F(x,y)=16 x=2.5 y=1.8 1 F(x,y)=14 x=3 y=1 F(x,y)=15 x=2 y=2.1 2.1 F(x,y)=14.8 x=2 y=2 2 F(x,y)=14.5 x=0.8 y=3 2.2 a) Write the constraint added at each branch. b) Have we reached the optimal solution? If so, explain why, else say from which node we should branch and with which constraints. 41

3. Let us consider an integer linear programming problem for which the following data of its branch-and-bound tree are known: F(x,y)=3832.5 x=191.6 F=3840 x=192 1 F=3838 x=190.5 y=1 F=3848 x=191 y=1 0 3 5 2 F=3835.5 x=191 y=0.5 4 F=3838 x=191 y=0 6 F=3840.4 x=190 y=1.4 a) Reason out if it is a maximization or minimization problem. b) Have we reached the optimal solution? If we have, explain why, else say from which node we should branch and with which constraints. c) Write the problem solved in node 6. 4. Given a certain mixed integer linear programming problem, where variables x and z have integrality condition, with the following branch-and-bound tree: F(x,y,z)=41.36 x=0.05 y=8 z=8.58 0 F(x,y,z)=41.33 x=0 y=8 z=8.67 1 2 F(x,y,z)=26.4 x=1 y=4.4 z=4.6 F(x,y,z)=39 x=0 y=7.67 z=8 1.1 Infactible 1.2 a) Reason out if it is a maximization or minimization problem. b) Have we reached the optimal solution? If we have, explain why, else say from which node we should branch and with which constraints. c) Write the problem solved in node 1.2. 42

5. Let us consider an ILP problem with two variables x and y. Let us suppose we have the following associated branch-and-bound tree: a) Reason out if it is a maximization or minimization problem. b) Write the constraints that have been added at each branch. c) Reason out if the optimal solution has already been found. If not, branch from the appropriate node. d) If the original problem was a mixed ILP one where only variable x had to be integer, what would be the answer to question (c)? 6. Let us consider an ILP problem with two variables x and y. Let us suppose we have the following associated branch-and-bound tree: a) Reason out if it is a maximization or minimization problem. b) Write the constraints that have been added at each branch. c) Reason out if the optimal solution has already been found. If not, branch from the appropriate node. d) If the original problem was a mixed ILP one where only variable x had to be integer, what would be the answer to question (c)? 43

7. A linear problem with three integer variables has the following associated branch-and-bound tree: Linear relaxation (x,y,z)=(3 4, 0, 3 8) O.F.=25 4 Problem 1: (x,y,z)=(3,0,4) O.F.=25 Problem 2: (x,y,z)=(11/3, 0, 3) O.F.=23 a) Write the constraint that has been added at each branch. b) Check the right option: The optimal solution cannot be obtained yet: we have to branch on node 2. The optimal solution is (3, 0, 4) with O.F.=25 The optimal solution is (11/3, 0, 3) with O.F.=23 The optimal solution is (4, 0, 4) with O.F. =28 8. The following solutions have been obtained in an integer linear programming problem: L.R. (x 1,x 2,x 3 )=(4, 3'5, 2) F=53'5 1 (x 1,x 2,x 3 )=(4'5, 4, 0) F=51'5 2 (x 1,x 2,x 3 )=(4, 3, 1'5) F=49 1.1 Infeasible 1.2 (x 1,x 2,x 3 )=(4, 4, 0) F=48 2.1 (x 1,x 2,x 3 )=(4, 2 5, 1) F=44'5 2.2 (x 1,x 2,x 3 )=(3 5, 3, 2) F=47'5 a) Write the constraints that have been added at each branch. b) Reason out if the optimal solution has been found. If it has been, say which one it is, else say form which problem we should branch and with which constraints. 44

9. Let us consider the following mixed integer programming problem: Min. F = 3x 1 + 2x 2 + 5x 3 s. t. 2x 1 + x 2 + 2x 3 12 x 1 + x 2 + 4x 3 8 x 1, x 2, x 3 0 x 3 integer If the first problem solved when applying the branch-and-bound method provides the optimal solution ( ) 3 (16 3,0, 2 3), with F = 58/3, check x 1,x2,x = the box(es) that make sense for the solutions of the subproblems obtained after branching for the first time: x 1,x,x = with F=20 5 and ( ) 1,x2,x3 = (5, 1, 1 2) x with F=20 and ( ) 1,x2,x3 = (5, 0, 1) x with F=18 and ( ) 1,x2,x3 = (4, 1, 1) x,x,x with F=22 and (,x,x3) (6, 1, 0) ( 2 3 ) (6, 0, 1 2) ( ) 1,x2,x3 = (4,4,0) ( ) 1,x2,x3 = (6, 0, 0) ( ) 3 (3, 4,1) 1 2 = x with F=19 5 x with F=20 x with F=19 x with F=22 1 2 = 10. Let us consider the following branch-and-bound tree corresponding to an integer linear programming problem: (x,y,z)=(16/3, 0, 2/3), F=58/3 (x,y,z)=(5, 1, 1/2), F=19,5 (x,y,z)=(6, 0, 1/2), F=20,5 (x,y,z)=(4, 4, 0), F=20 (x,y,z)=(5, 0, 1), F=20 (x,y,z)=(6, 0, 1), F=23 (x,y,z)=(5, 2, 0), F=22 a) Add the constraint corresponding to each branch. b) Reason out what the optimal solution is, and if it is a global maximum or minimum. c) Determine which of the seven solutions seen in the tree is wrong and why. 45